Hundreds of thousands of volunteers have helped to overturn almost a century of galaxy classification, in a new study using data from the longstanding Galaxy Zoo project. The new investigation uses classifications of over 6000 galaxies to reveal that ‘well known’ correlations between different features are not found in this large and complete sample. (1)
By encoding their specialized knowledge into a computer game, researchers enabled citizen scientists to successfully design synthetic proteins for the first time. (2)
Science was once upon a time something for the elite few. Now it is a matter of everyone. Science was once upon a time related to wisdom. Now it is related to date analysis. Science was once upon a time part of our belief in God. Now we just believe in us. We used to be part of God. Knowing everything by bring part of it. Now we observe the million pieces we have created. At the end we will know everything. But not everything that there is. But everything that we want them to be. For we are not actually observing anything. But we have set up mirrors. To observe our selves… Through the looking glass…
Abstract: The Sun being another star is common knowledge. However, as it happens with all things that are considered obvious, few can actually name exactly who and how the Sun was considered as just another star. It seems that besides evidence from stellar spectroscopy and the measurement of astronomical distances, philosophical principles also played a major role in the building of this knowledge. From the De l’infinito universon e mondi (On the Infinite Universe and Worlds) of Bruno in 1584 up to the Principia Philosophiae of Rene Descartes in 1644, people had started adhering to the idea of the Sun being nothing more than a common star. This idea – also enhanced by the ideas of Copernicus – was later on verified by spectral data and since the era of the Jesuit priest and astronomer Angelo Secchi it is considered an established fact nowadays. However the current huge gaps of our understanding on the nature of the universe call for being much more careful when calling any such knowledge a ‘fact’. More humility is highly advised, especially in a sector of knowledge where we have recently realized we can only account for only the ~5% of it. At the end, acknowledging our limitations is far more important than projecting our beliefs…
[Greek abstract can be found at the end of the article]
1. A question posed…
Once upon a time I had a discussion with friends on cosmology. There, among other things, the question of what are the stars came up. And it was very interesting for me to acknowledge that the answer to this seemingly simple question is not so simple after all…
So what are the stars?
How do we know our Sun is a star?
Who discovered that the stars are “Suns”?
To answer this we must first travel many centuries back and delve into philosophy and the history of science. There, we will find long forgotten assumptions that still dictate how we think about the cosmos.
At the end, you need not worry about the stars not being stars.
They could be, or they could be not – at the end it matters not.
What matters is the human tendency of clinging to dogmas for thousands of years without a single hint of remorse. And this is something we should certainly look into and fix if we are – ever – to unlock the mysteries of the cosmos around us…
2. Searching for an answer…
The first thing to do when you have a question is to search for an answer. But the answer to the question “How do we know the Sun is like the stars?” is not easy to find.
The questions seems to most people so obvious (and perhaps stupid) that they do not even care of explaining why we consider the Sun a star (or vice versa). When this is asked they most usually answer with a simple “Yes they are the same, end of story” attitude that leaves little room of questions, unless of course you want to be ridiculed online that you are unaware of basic astronomy that even kids in the kindergarden know.
Let us look together some of the answers found online for the matter…
2.1 Answers that are not clear answers…
Below I document what various resource in the Internet have to say on the nature of the Sun and the stars. They show clearly the main problem: When something is considered obvious, little effort is put into explaining it. And it is in these ‘obvious’ things that the problems usually arise…
Let us see some excerpts from these resources below…
Who determined that the sun was a star, like the stars in the nighttime sky? Answer: No single astronomer had this realization. Prominent thinkers considered the possibility since classical antiquity; they had creative rhetorical argument on their side, but no proof. By the late 19th century, we knew what stars were, and we knew the distances from the earth to a few stars and to the sun; with that data, astronomers determined that these bodies released energy in roughly comparable amounts. Then spectroscopic examination revealed that the chemical elements in the solar atmosphere were just like those found in common yellow-colored stars spread across the sky. (David H. DeVorkin, senior curator, National Air and Space Museum) [3]
In other words: That the Sun is a star and vice-versa we know because we… know.
Q: im having a hard time believing that the stars are really suns. So from a stars distance, does our sun look like a tiny little star? – Christine (age 16) A: Yes. [4]
In other words: Don’t ask. The stars are suns. And it is a sin to question that.
Given that the only observational information we have on stars is the light we receive, you might think there isn’t much we can learn about them. But by comparing positions, brightness, and spectra over time, and comparing these with observations of our own star, the Sun, we can actually create accurate models that explain and predict stellar characteristics and behavior. [1]
In other words: We use the comparison between the Sun and the stars to draw conclusions. But what about the Sun being a star? Is that something we consider valid because of some specific reason? Again, this “knowledge” is implied but not specifically mentioned (let alone proved).
As astronomers gaze into the depths of space, they do so with unease: They don’t know precisely what the universe is made of. It’s not just the true nature of dark matter that eludes them; so does the essence of the stars that speckle the sky and populate the many galaxies throughout the cosmos. Surprisingly, no one knows the stars’ exact chemical composition: how many carbon, nitrogen and oxygen atoms they have relative to hydrogen, the most common element. These numbers are crucial, because they affect how stars live and die, what types of planets form and even how readily life might arise on other worlds. [5]
In other words: We do not know many things about stars or the universe. (keep that in the back of your head, we will use it again)
William Herschel was the first astronomer to attempt to determine the distribution of stars in the sky. During the 1780s, he established a series of gauges in 600 directions and counted the stars observed along each line of sight. From this he deduced that the number of stars steadily increased toward one side of the sky, in the direction of the Milky Way core. His son John Herschel repeated this study in the southern hemisphere and found a corresponding increase in the same direction.[30] In addition to his other accomplishments, William Herschel is also noted for his discovery that some stars do not merely lie along the same line of sight, but are also physical companions that form binary star systems. [6]
In other words: Here is the first specific mention of something concrete. Someone did measure something regarding the stars and drew a specific conclusion. Of course the only thing he was based upon was what he saw: the light of the stars. (keep that also in mind)
The science of stellar spectroscopy was pioneered by Joseph von Fraunhofer and Angelo Secchi. By comparing the spectra of stars such as Sirius to the Sun, they found differences in the strength and number of their absorption lines—the dark lines in stellar spectra caused by the atmosphere’s absorption of specific frequencies. In 1865, Secchi began classifying stars into spectral types.[31] However, the modern version of the stellar classification scheme was developed by Annie J. Cannon during the 1900s. [6]
In other words: Here we have another specific example of concrete science. We measure something and compare the data we have for the Sun and other stars (given that the other stars are suns of course). Spectroscopy is a big leap towards understanding the stars and their nature, since it can provide many data for the properties of these celestial objects. It is only a pitty that this is all we have, along with distance measurements. What else could we have anyway? We have never gone to the stars, we have only approached somehow our own star.
Various Quora questions (Q) and answers (A) can also be found below:
Q: Are the stars we see in the sky actually Suns from other solar systems? – A: Yes – our Sun is just another “star” and those stars are really “suns”. Same exact thing. [7]
Q: Are the night sky stars all suns? A: Yes, almost all of what we see with our eyes in the night sky as ‘stars’, are actual stars (or suns, as it was stated in the question, presumably to avoid using the word ‘stars’ twice, with different meanings). [8]
In other words: Yes, the stars are like the sun. And it is obvious.
The Sun is the dominant object in the solar system by mass and total energy content. The irradiance of the Sun drives climate on the planets and is the primary source of energy for the biosphere of the earth. The Sun is a Rosetta Stone for the study of astrophysical processes at resolutions that cannot be easily attained for other stars. The results of these solar studies can be applied toward an understanding of other stars, including the properties of their atmospheres and interior structures. In the realm of physics the Sun plays a unique role. The element helium—the second most abundant element in the universe after hydrogen—was discovered in the solar spectrum. The Sun serves as among the test beds for Einstein’s theory of General Relativity. The nature of subatomic particles called neutrinos—the byproducts of nuclear reactions in the hot and dense core of the Sun and sun-like stars—was elucidated as a result of solar investigations. The Sun serves as a laboratory for the study of plasma physics, i.e., the study of the interactions between ionized gas and magnetic fields. [9]
In other words: The Sun is used as a reference to induce assumptions regarding the other stars. So the assumption that the Sun is like the other stars is even more important that we might have thought: The stars are used to draw conclusions for our Sun and the Sun is used to study better the other stars.
ELI5:How do we know that stars are suns? Answer 1: Through spectroscopy, we can determine the composition of stars through their emissions. The experiment you’re looking for is performing this spectroscopy on the sun, and on stars, and discovering that they have similar characteristics. Answer 2: I think to short answer is parallax and spectral analysis. Parallax is a small shift in relative position when the point of view changes. This can tell us the distance. Spectral analysis is a way of determining elemental make up because different sets of wavelengths of light are caused by different elements. [10]
In other words: Again the importance of spectroscopy is emphasized in determining the nature of the Sun and the stars. Indeed this is an excellent tool in analysing the temperature, the composition even the rotation of the celestial objects. Is it a perfect tool? Of course not. What tool is perfect? But it is a very scientific and credible tool in giving us insight in these fascinating objects that linger in the night sky…
So is this the answer we were looking for?
Is spectroscopy the answer to why we consider the Sun another star?
It seems so, yes.
Even though most resources do not mention it clearly, it is evident that similarities of the spectra of the Sun and the stars have made scientists figure out that they must be similar objects. However this answer should not satisfy the researcher here.
Is such a similarity enough?
Looking more into the subject we will discover that there are additional elements needed in order to accept that the sun and the stars are one and the same thing…
3. Regarding spectral analysis
A small parenthesis regarding the spectral analysis based on which we deduce the similarity between the Sun and the stars is needed here.
Electromagnetic radiation with the shortest wavelengths, no longer than 0.01 nanometer, is categorized as gamma rays. Electromagnetic radiation with wavelengths between 0.01 nanometer and 20 nanometers is referred to as X-rays. Radiation intermediate between X-rays and visible light is ultraviolet (meaning higher energy than violet). Electromagnetic radiation with wavelengths between roughly 400 and 700 nm is called visible light because these are the waves that human vision can perceive. Between visible light and radio waves are the wavelengths of infrared or heat radiation. After infrared comes the familiar microwave, used in short-wave communication and microwave ovens. All electromagnetic waves longer than microwaves are called radio waves, but this is so broad a category that we generally divide it into several subsections. [43]
Looking into the light coming to us from the sky, we can deduce many information. Essentially by using the emission or absorption spectra we can conclude things regarding the composition of the stars or the atmosphere of planets, their temperature, density, mass, radius, distance, luminosity, and relative motion [35] [40].
In 1860 Gustav Kirchhoff proposed the idea of a black body, a material that emits electromagnetic radiation at all wavelengths. In 1894 Wilhelm Wien derived an expression relating the temperature (T) of a black body to its peak emission wavelength (λmax). [35]
This equation is called Wien’s Law. By measuring the peak wavelength of a star, the surface temperature can be determined. For example, if the peak wavelength of a star is 502 nm the corresponding temperature will be 5778 kelvins. [35] An object at a higher temperature emits more power at all wavelengths than does a cooler one. In a hot gas, for example, the atoms have more collisions and give off more energy. In the real world of stars, this means that hotter stars give off more energy at every wavelength than do cooler stars [43].
Figure: Radiation Laws Illustrated. This graph shows in arbitrary units how many photons are given off at each wavelength for objects at four different temperatures. The wavelengths corresponding to visible light are shown by the colored bands. Note that at hotter temperatures, more energy (in the form of photons) is emitted at all wavelengths. The higher the temperature, the shorter the wavelength at which the peak amount of energy is radiated (this is known as Wien’s law). [43]
Also the higher the temperature, the shorter the wavelength at which the maximum power is emitted [43] (see figure above).
So that could be a way to distinguish between planets and stars.
The study of many thousands of stellar spectra in the late Nineteenth Century led to the development of our modern classification system for stars [37].
However note that there are also hot planets at the size of Jupiter that have a temperature between an Earth-sized planet and a star. These planets reside in an intermediate section and called for corrections in the models used to analyze spectra [36].
Also note that there are other objects rthat are not exactly stars and which give out similar (but different) spectra, like the quasars or some types of exotic stars [37].
Credit: 2dF Quasar Survey Characteristic QSO spectrum showing distinct, strong, redshifted emission lines of a quasar. [37]
These are examples that simply make the problem of accepting the similarity of the Sun with the other stars also more of a definition (and, thus, philosophical) one. What is the cut-off point beyond which we decide that a celestial object is ‘different’ than another?
It seems that searching into the philosophy behind modern cosmology we can find out more factors that weighted in the acceptance of the ‘stars-sun analogy’.
4. Assumptions in modern astronomy
Two basic assumptions of today’s cosmology are the homogeneity and isotropy [25] of the universe, something also known as the Cosmological Principle [24] [26] [27].
Also related to that principle is the Copernican principle, a principle on which many articles have been written in Harmonia Philosophica. You can see the “Earth at the center of the universe?” article for more on that.
Essentially the Copernican principle postulates that humans are not in any way in a priviledged position in the cosmos. As already said this is connected with the notion of isotropy in the cosmos (the Cosmological principle) in various ways: If we are in a non-priviledged position then we are not seeing anything ‘different’ in any direction and, vice-versa, if we do not look anything ‘different in any direction we are not in a priviledged position.
Let us now start a journey in the history behind the acceptance of the Sun as being an ordinary star to see how philosophy had also a thing or two to say regarding what we believe about our star…
5. The role of philosophy in accepting that the Sun is a star
In everything, our philosophy plays a very important and crucial role in what we say, decide and believe. Astronomy and cosmology is not an exception.
The above-mentioned assumptions guided astronomers throughout the recent centuries in deriving conclusions for the stars, as much as observational data did.
To acknowledge this is important not only because it allows us a better understanding of the way science works, but also because it may help us avoid prejudiced thinking in the future.
5.1 A world of ‘worlds’: The philosophers speak of the Sun as a ‘star’
What is a world according to modern throught?
As a result of shifting views of the universe the very idea of “world” (in Latin, Mundi) was changing. In the Aristotelian cosmos, the world was effectively synonymous with the Earth. The sphere of the world and the terrestrial realm were one in the same [29].
In part, what we think to-day when we think of a ‘word’ is based on the ideas of the Dominican Friar Giordano Bruno (an Italian philosopher who lived from 1548 to 1600) who published De l’infinito universon e mondi (On the Infinite Universe and Worlds), in 1584. As part of a suite of mystical, magical and heretical ideas and in part, spurred on by Copernicus’s ideas, he suggested that Earth was one of many inhabited worlds in an infinite universe and that the stars were suns, which had their own worlds [29].
Giordano Bruno decided that if the Earth is a planet just like the others, then it does not make sense to divide the Universe into a sphere of fixed stars and a solar system. He said that the Sun is a star (i.e. not anything special), that the Universe is infinitely large, and that there are many worlds. He was condemned by both the Roman Catholic and Reformed Churches for this as well as other things and was burnt alive in Rome in 1600 for heresy [2]. As he wrote at the time: “The composition of our own star and world is the same as that of as many other stars and worlds as we can see” [13].
In other words, it seemed reasonable to him that the Sun was merely another star, and he subsequently made a distinction between “suns” which generate their own light and heat; and the “earths” and moons which revolve and are nourished and powered by them. One esteemed modern astrophysicist, Steven Soter, has even suggested that Bruno was the first person in history to truly grasp the concept that “stars are other suns with their own planets” [13].
Once Earth became one planet among many orbiting the sun, those planets became Earth like worlds. This new understanding of worlds is reflected in the title of The Discovery of a World in the Moon from the 1630’s [29]. But as it took a long time for the Copernican model of the cosmos to win out over competing models, it took a considerable bit of time for ideas similar to Bruno’s to come to fruition. [29]
The overlapping circles in Tycho Brahe’s geocentric model of the cosmos created a significant problem for the Aristotelian notions of the heavenly spheres. If Brahe was right and the orbits of the planets crossed each other each other then they couldn’t be a set of solid. Rene Descartes offered a solution to this problem in his 1644 Principia Philosophiae. In Descartes system, like Aristotle’s, the universe was full of matter, there was no such thing as empty space. To explain motion Descartes introduced the concept of vortices. The system consisted of different kinds of mater or elements rubbing up against each other. His model included three different kinds of elements: luminous, transparent, and opaque. Luminous was the smallest and was what the stars were made of. Earth and the planets were made up of the denser opaque. The space between the planets and the stars was made up of transparent He stated that Lumnious would settle at the center of these vortices and the transparent and opaque elements would keep shifting around each other. This shifting created the movement of objects in the heavens. [30]
The increasing acceptance of Descartes theory of vortices in the later half of the 17th century brought with it the idea that the stars were like our sun and had their own planets orbiting around them. Bernard le Bovier de Fontenelle’s popular 1686 book Entretriens sur la pluralite des mondes (Conversations on the plurality of worlds) broadly disseminated this notion, in a range of editions and translations. (You can read a full-digitized copy External of an 1803 English translation of Conversations on the Plurality of Worlds online from the Library of Congress collections) [29].
The book Conversations on the Plurality of Worlds presents fictional discussions between a philosopher and his hostess, a marquise. As the two characters walk the grounds of her garden at night they discuss the stars above them. Their conversations touch on the features of the Copernican system, potential encounters with extraterrestrial life and the idea of the universe as a boundless expanse. As the book was translated into a variety of languages and republished in new editions for hundreds of years, it presented both this cosmology and the idea of life on other worlds to a range of audiences [29].
Changing ideas about the structure of the universe are also well illustrated in diagrams from William Derham’s 1715 book Astro-Theology. Derham, an English natural philosopher, astronomer and clergymen wrote a series of works exploring connections between natural history and theology. From his perspective, the shift to thinking about the plurality of worlds was significant enough that it should be set alongside the Copernican Revolution as one of the three major shifts in thinking about the nature of the universe [29].
The above history provides a good description on how philosophy and most importantly our idea of us being insignificant (part of a larger universe where everything is the same everywhere – a.k.a. Cosmological principle) like postulated by the Copernical principle dictated our journey towards understanding the cosmos.
The next steps came from ‘science’, in the modern sense of the word…
5.2 The advent of spectroscopy: The scientists speak of the Sun as a ‘star’
In 1666, Isaac Newton showed that a prism separated white light into a spectrum of its constituent parts, rather than creating the rainbow colors that are seen. In 1802, William Wollaston then constructed a spectrometer which showed the Sun’s spectrum on a screen, but noted that there were dark bands of missing colors [13].
In 1814, Joseph von Fraunhofer invented the spectroscope and mapped 574 of these lines, after which a number of scientists helped advance the study of spectroscopy, including Gustav Kirchhoff and Robert Bunsen who in 1857 were able to establish a connection between chemical elements and their own individual spectral patterns. [13]
Further study revealed that each element absorbs light of a particular color, thus leaving a specific “signature” line. And after spectroscopes were coupled to telescopes, scientist were able to identify additional chemical elements, and work our the chemical composition of the stars, as well as distinguish between nebulae and galaxies in the night sky. [13]
During this period, an Italian Jesuit priest and astronomer, Angelo Secchi (1818-1878), became a pioneer in the study of stellar spectroscopy, and through analysis of some 4,000 stellar spectrograms discovered that the stars came in a limited variety of types distinguishable by their unique spectral patterns. He subsequently devised the first stellar classification system, and is recognized as being one of the first scientists to definitively state that the Sun is a star [13].
So there you go.
What philosophers postulated centuries ago, was now verified by scientists. Strange how we always verify things we already know, is it not?
But again, could we even know what we do not?
Note: Search Harmonia Philosophica for more on the limits of knowledge, the limits of science and the limits of sensing the cosmos.
Instead of Conclusion: Things we do not know…
But what do we know exactly?
In astronomy the things we do not know are much more than the things we do. Cosmology is full of mysteries that are bravely admitted by astronomers are big gaps in our understanding of the cosmos.
Examples of things that we do not know include:
How does universe’s inflation work [32].
How singularities form and what they actually are [32].
How galaxies or stars formed [32].
Ultra-energetic cosmic rays [33].
Dark matter/ Dark energy [33] [34].
The Pioneer anomaly [33].
The Wow signal [33].
Details about massive stars: How far away they are, how they form, what is their maximum mass etc. [31]
Missing baryons [34].
How do stars explode [34].
Why is the sun’s corona so hot? [34]
All in all, even the example of dark matter and dark energy that account for more than ~95% of the cosmos [45] is enough for someone to understand that our knowledge is too much limited for us to draw safe conclusions about the cosmos around us.
So what is the Sun?
What are the stars?
Sure, stellar spectroscopy gives us many indications of the answer. But this is the only thing it provides: indications. We cannot know know unless we see. There are many other objects in the sky with similar yet different spectra, like quasars for example.
What is more one of the basic elements used for the classification and study of stars – distance – is measured by methods that have limitations (look at the Appendix I – Stellar parallax for more on that).
Put these notes together along with our knowledge gaps mentioned above, and you will see that our understanding of the Sun as a star (or the stars as Suns) is so safe as the understanding of a Neanderthal gazing the Sun millions of years ago.
Surely the stars may be sun and they most probably are. But our current knowledge of that is based as much in philosophy as it is based on scientific evidence from spectroscopy.
It is important to not only understand but also acknowledge that. Because through that acknowledgement we will understand not the stars but our own self here on Earth better.
At the end it is not about what is or what is not.
It is not about the Sun is a star or not.
All that matters is that we can think in the dark.
Without anything. Within everything.
Admiring the cosmos without need to categorize.
Because, as Shestov once said, when we try to categorize and understand things we just break them down into pieces that fit into the little boxes we have in our mind. (Shestov also wrote many interesting notes regarding astronomy and astrology, search Harmonia Philosophica for them)
[25] K. Migkas, G. Schellenberger, T. H. Reiprich, F. Pacaud, M. E. Ramos-Ceja, L. Lovisari. Probing cosmic isotropy with a new X-ray galaxy cluster sample through the LX–T scaling relation. Astronomy & Astrophysics, 2020; 636: A15 DOI: 10.1051/0004-6361/201936602
Ελληνική Σύνοψις (Greek abstract): Το ότι ο Ήλιος είναι ένα απλό αστέρι είναι κοινή γνώση. Ωστόσο, όπως συμβαίνει με όλα τα πράγματα που θεωρούνται προφανή, λίγοι μπορούν στην πραγματικότητα να ονομάσουν ακριβώς ποιος το διατύπωσε πρώτος και πως και γιατί ο Ήλιος θεωρήθηκε αστέρι. Φαίνεται ότι εκτός από στοιχεία από την αστρική φασματοσκοπία και τη μέτρηση αστρονομικών αποστάσεων, ορισμένες φιλοσοφικές αρχές έπαιξαν επίσης σημαντικό ρόλο στην οικοδόμηση αυτής της γνώσης. Από το De l’infinito universon e mondi (On the Infinite Universe and Worlds) του Μπρούνο το 1584 έως το Principia Philosophiae of Rene Descartes το 1644, οι άνθρωποι είχαν αρχίσει να συνηθίζουν στην ιδέα ότι ο Ήλιος δεν είναι τίποτα περισσότερο από ένα κοινό αστέρι. Αυτή η ιδέα – ενισχυμένη και από τις ιδέες του Κοπέρνικου – επαληθεύτηκε αργότερα με τα φασματικά δεδομένα και από την εποχή του Ιησουίτη ιερέα και αστρονόμου Angelo Secchi θεωρείται ως καθιερωμένο γεγονός στις μέρες μας. Ωστόσο, τα σημερινά τεράστια κενά στην κατανόησή μας σχετικά με τη φύση του σύμπαντος απαιτούν να είμαστε πολύ πιο προσεκτικοί όταν αποκαλούμε τέτοια γνώση ως «γεγονός». Συνιστάται περισσότερη ταπεινοφροσύνη, ειδικά σε έναν τομέα της γνώσης όπου έχουμε πρόσφατα συνειδητοποιήσει ότι μπορούμε να μιλήσουμε μόνο το ~ 5% αυτής. Εν τέλει, το να αναγνωρίζουμε τα όρια μας είναι πολύ σημαντικότερο από το να προβάλλουμε τα πιστεύω μας…
APPENDIX I – Stellar parallax
Stellar parallax is the apparent shift of position of any nearby star (or other object) against the background of distant objects. Created by the different orbital positions of Earth, the extremely small observed shift is largest at time intervals of about six months, when Earth arrives at opposite sides of the Sun in its orbit, giving a baseline distance of about two astronomical units between observations. The parallax itself is considered to be half of this maximum, about equivalent to the observational shift that would occur due to the different positions of Earth and the Sun, a baseline of one astronomical unit (AU) [22].
The first known astronomical measurement using parallax is thought to have occurred in 189 B.C., when a Greek astronomer, Hipparchus, used observations of a solar eclipse from two different locations to measure the distance to the moon [23].
Limitations of Distance Measurement Using Stellar Parallax
Parallax angles of less than 0.01 arcsec are very difficult to measure from Earth because of the effects of the Earth’s atmosphere. This limits Earth based telescopes to measuring the distances to stars about 1/0.01 or 100 parsecs away. Space based telescopes can get accuracy to 0.001, which has increased the number of stars whose distance could be measured with this method. However, most stars even in our own galaxy are much further away than 1000 parsecs, since the Milky Way is about 30,000 parsecs across. The next section describes how astronomers measure distances to more distant objects. [11]
Although it is correct to take account of the relative motion of the solar system and the star being measured, in fact the trigonometric parallax method is limited to measurements of relatively nearby stars, so this relative motion is rather small. [12]
Astrometric surveys such as Gaia and LSST will measure parallaxes for hundreds of millions of stars. Yet they will not measure a single distance. Rather, a distance must be estimated from a parallax. In this didactic article, I show that doing this is not trivial once the fractional parallax error is larger than about 20%, which will be the case for about 80% of stars in the Gaia catalogue. Estimating distances is an inference problem in which the use of prior assumptions is unavoidable. I investigate the properties and performance of various priors and examine their implications. A supposed uninformative uniform prior in distance is shown to give very poor distance estimates (large bias and variance). Any prior with a sharp cut-off at some distance has similar problems. The choice of prior depends on the information one has available – and is willing to use – concerning, for example, the survey and the Galaxy. I demonstrate that a simple prior which decreases asymptotically to zero at infinite distance has good performance, accommodates non-positive parallaxes, and does not require a bias correction. [14]
Calculating the distance to the object is easy since the parallax calculations are based on simple trigonometry, although the triangles found in parallax measurements have no relation to those found in “normal” trigonometry. In the picture at the top of the page, the distance that the star appears to have moved when viewed from different perspectives represents its distance. However, even at a relatively close distance, such as that of Proxima Centauri, which is only 4.2 light years away, this angle is extremely small. In fact, Proxima Centauri has a parallax angle of only 0.7687 ± 0.0003 seconds of arc, which roughly equates to an angle that subtends an object 2 cm across, but seen from a distance of 5.3 km away. As distances to objects increase, parallax angles get progressively smaller, until they become so small that they are impossible to measure, even with the most sophisticated equipment available today, and it is at this point that discrepancies in the distance/luminosity stats for objects arise.
The problem of negative parallax
The parallaxes of distant stars should be zero (or at least indistinguishable from zero). If the parallaxes have an uncertainty (which they do), then half of the parallaxes of distant stars will be negative. I think this is all that you are finding in the case of absolute Hipparcos parallaxes. The quote you give from the 1943 paper is talking about relative parallaxes. When you determine relative parallax you find the apparent movement in the sky with respect to a bunch of comparision stars in the same region. You make the assumption that most of these stars are very far away and have zero parallax. If a large fraction of the stars in fact have a positive and large parallax (because you are looking towards a nearby cluster), then the relative parallaxes of the genuinely distant stars in the cluster can end up negative on average. [16]
How should we handle negative parallax?
For closely aligned sources (separated by 0.2–0.3 arcsec), which are only occasionally resolved in the Gaia observations, confusion in the observation-to-source matching can lead to spurious parallax values which are either very large or have a negative value very far away from zero in terms of the formal parallax uncertainty quoted in the catalogue. These sources tend to be faint and located in crowded regions and are also associated with unreliable (large) proper motions (Gaia Collaboration et al. 2018b). Guidance on how to clean samples from spurious parallax values is provided in Lindegren et al. (2018). [17]
The systematic errors in the parallaxes are estimated to be below the 0.1 mas level (Lindegren et al. 2018) but the following systematics remain. There is an overall parallax zeropoint which, from an examination of QSO parallaxes, is estimated to be around −0.03 mas (in the sense of the Gaia DR2 parallaxes being too small). The estimated parallax zeropoint depends on the sample of sources examined (Arenou et al. 2018) and the value above should not be used to ’correct’ the catalogue parallax values. [17]
It depends how negative the parallax is and what your “prior” knowledge is of the distance to the star is. As another answer suggests, there are some spurious large negative (and positive) parallaxes for faint, crowded sources. If possible, these should be removed. If this is not the case, and the parallax is negative, but close to zero within its uncertainty, then it is telling you that you have a lower limit to the distance of the object (i.e. measurement uncertainties have led to a small negative parallax). Crudely speaking, you could add a couple of error bars to the parallax and treat that as a 95% upper limit (remember the 0.1 mas possible systematic error too). Under no circumstances should you “use them as is”, since there is no physical basis for a negative parallax or negative distance. [17]
Although should not use the negative parallaxes, you should not ignore them either. If you are looking at populations of objects, removing those with negative parallaxes will lead to significant bias in your results, as Luri et al. 2018 [18] has shown. [17]
Negative parallaxes can be interpreted as the observer (in this case Gaia satellite) going the “wrong way around the sun” as mentioned in this Jupyter Notebook by Anthony Brown. [17]
Any astrometric catalogue that lists parallaxes will contain negative parallaxes, which at first sight appear physically implausible, yet they are an entirely valid measurement of the true positive parallax in the presence of (large) noise. This notebook discusses how negative parallaxes may arise even for “perfect” measurements (suffering only from random Gaussian noise, without any systematic errors). [19]
The reason a parallax can turn up negative is simple. Errors can cause a star position to off by any direction. During the six months we measure the parallax, we expect the star’s position to shift from A to B. In this case, the true parallax was about the same size as a typical error, but the first error pushed the position reading to roughly where B is, and the second error pushed the star to roughly where A is. So the star appears to move from B to A instead of A to B: the parallax is the wrong way round. We can’t know what the error was, so we can’t subtract those. [20]
Other papers also emphasize that negative parallaxes are simply errors [21].
At the end, everything could be just a problem with the assumptions on which stars are “background stars”: Stellar parallax is the apparent motion of stars relative to other stars, which also have parallaxes. We do not know beforehand which stars are closer than the others, and these have to be inferred using statistical analysis from the entire data. The parallaxes of distant stars should be practically zero. And because they have a statistical uncertainty, then half of these near-zero parallaxes will be negative. [28]
Of course again this places the negative parallax in the category of an ‘error’, thus dismissing it altogether. It is like saying ‘If you have a negative parallax, then your measurement is wrong’.
All in all, the phenomenon deserves more attention and perhaps the advances in astronomy will soon provide a more definite answer on the problem of astronomical distance measurements.
APPENDIX II – Astronomical spectroscopy
Astronomical spectroscopy is the study of astronomy using the techniques of spectroscopy to measure the spectrum of electromagnetic radiation, including visible light and radio, which radiates from stars and other celestial objects. A stellar spectrum can reveal many properties of stars, such as their chemical composition, temperature, density, mass, distance, luminosity, radius and relative motion using Doppler shift measurements. Spectroscopy is also used to study the physical properties of many other types of celestial objects such as planets, nebulae, galaxies, and active galactic nuclei. [35] [40]
Electromagnetic radiation with the shortest wavelengths, no longer than 0.01 nanometer, is categorized as gamma rays. Electromagnetic radiation with wavelengths between 0.01 nanometer and 20 nanometers is referred to as X-rays. Radiation intermediate between X-rays and visible light is ultraviolet (meaning higher energy than violet). Electromagnetic radiation with wavelengths between roughly 400 and 700 nm is called visible light because these are the waves that human vision can perceive. Between visible light and radio waves are the wavelengths of infrared or heat radiation. After infrared comes the familiar microwave, used in short-wave communication and microwave ovens. All electromagnetic waves longer than microwaves are called radio waves, but this is so broad a category that we generally divide it into several subsections. [43]
Different celestial objects produce different types of spectra. The spectrum of an object is one means of identifying what type of object it is. How different spectra arise is shown in the schematic diagram below. [37]
Credit: Adapted from a diagram by James B. Kaler, in “Stars and their Spectra,” Cambridge University Press, 1989.: How continuous, emission and absorption spectra can be produced from same source. [37]
Continuum spectrum: In this diagram, a dense hot object such as the core of a star acts like a black body radiator. If we were able to view the light from this source directly without any intervening matter then the resultant spectrum would appear to be a continuum as shown bottom left in the figure above. [37] [38]
Absorption spectrum: Most stars are surrounded by outer layers of gas that are less dense than the core. The photons emitted from the core cover all frequencies (and energies). Photons of specific frequency can be absorbed by electrons in the diffuse outer layer of gas, causing the electron to change energy levels. Eventually the electron will de-excite and jump down to a lower energy level, emitting a new photon of specific frequency. The direction of this re-emission however is random so the chances of it travelling in the same path as the original incident photon is very small. The net effect of this is that the intensity of light at the wavelength of that photon will be less in the direction of an observer. This means that the resultant spectrum will show dark absorption lines or a decrease in intensity as shown in the dips in the absorption spectrum top right in the diagram above. Stellar spectra typically look like this. [37]
Emission spectrum: A third possibility occurs if an observer is not looking directly at a hot black body source but instead at a diffuse cloud of gas that is not a black body. If this cloud can be excited by a nearby source of energy such as hot, young stars or an active galactic nucleus then the electrons in atoms of the gas cloud can get excited. When they de-excite they emit photons of specific frequency and wavelength. As these photons can re emitted in any direction an external observer will detect light at these wavelengths. The spectrum formed is an emission or bright line spectrum, as shown by the middle spectrum in the figure above. [37]
Newton used a prism to split white light into a spectrum of color, and Fraunhofer’s high-quality prisms allowed scientists to see dark lines of an unknown origin. In the 1850s, Gustav Kirchhoff and Robert Bunsen described the phenomena behind these dark lines. Hot solid objects produce light with a continuous spectrum, hot gases emit light at specific wavelengths, and hot solid objects surrounded by cooler gases show a near-continuous spectrum with dark lines corresponding to the emission lines of the gases. By comparing the absorption lines of the Sun with emission spectra of known gases, the chemical composition of stars can be determined. [35]
An ideal thermal spectrum is shown on the left below. A spectrum of an actual star is shown on the right.
Stellar specturm example [44]
In addition to the continuous spectrum, a star’s spectrum includes a number of dark lines (absorption lines). Absorption lines are produced by atoms whose electrons absorb light at a specific wavelength, causing the electrons to move from a lower energy level to a higher one. This process removes some of the continuum being produced by the star and results in dark features in the spectrum [44].
In 1860 Gustav Kirchhoff proposed the idea of a black body, a material that emits electromagnetic radiation at all wavelengths. In 1894 Wilhelm Wien derived an expression relating the temperature (T) of a black body to its peak emission wavelength (λmax). [35]
This equation is called Wien’s Law. By measuring the peak wavelength of a star, the surface temperature can be determined. For example, if the peak wavelength of a star is 502 nm the corresponding temperature will be 5778 kelvins. [35] An object at a higher temperature emits more power at all wavelengths than does a cooler one. In a hot gas, for example, the atoms have more collisions and give off more energy. In the real world of stars, this means that hotter stars give off more energy at every wavelength than do cooler stars [43].
Figure: Radiation Laws Illustrated. This graph shows in arbitrary units how many photons are given off at each wavelength for objects at four different temperatures. The wavelengths corresponding to visible light are shown by the colored bands. Note that at hotter temperatures, more energy (in the form of photons) is emitted at all wavelengths. The higher the temperature, the shorter the wavelength at which the peak amount of energy is radiated (this is known as Wien’s law). [43]
By measuring the peak wavelength of a star, the surface temperature can be determined. For example, if the peak wavelength of a star is 502 nm the corresponding temperature will be 5778 kelvins. [35]
The spectra of galaxies look similar to stellar spectra, as they consist of the combined light of billions of stars. [35]
The reflected light of a planet contains absorption bands due to minerals in the rocks present for rocky bodies, or due to the elements and molecules present in the atmosphere. To date over 3,500 exoplanets have been discovered. These include so-called Hot Jupiters, as well as Earth-like planets. Using spectroscopy, compounds such as alkali metals, water vapor, carbon monoxide, carbon dioxide, and methane have all been discovered. [35]
Until recently, the modelling of the atmospheres of stars (e.g. Gray 2005) and that of the atmospheres of the Earth and other Solar system planets (e.g. Liou 2002) have developed largely independently. Models of stars applied to high-temperature objects with effective temperatures Teff > 3000K, with opacity dominated by the line and continuum absorption of atoms and atomic ions, whereas planetary atmosphere models applied to cool objects Teff∼ 100–300K, where the important processes were molecular absorption and scattering from molecules and cloud particles. [36]
This situation changed with the discovery in the mid-1990s of the first unambiguous brown dwarf, Gl 229B. (Nakajima et al. 1995; Oppenheimer et al. 1995) and the first hot Jupiter planets beginning with 51 Peg b (Mayor & Queloz 1995; Marcy et al. 1997). Many more such objects have now been discovered and reveal that planets and brown dwarfs populate an intermediate range of temperatures not explored previously. This has led to the requirement to develop new methods to model these atmospheres that cover the effective temperature range from below 1000K to more than 2000K. [36]
Spectroscopy is also used nowadays to detect exoplanets around distant stars [41] [42]. For example Transit spectroscopy is the ideal technique to probe temperate planets around M-dwarfs [38].
The idea of science being the best – or the only – way to reach the truth about our cosmos has been a major belief of modern civilization. Yet, science has grown tall on fragile legs of clay. Every scientific theory uses axioms and assumptions that by definition cannot be proved. This poses a serious limitation to the use of science as a tool to find the truth. The only way to search for the latter is to redefine the former to its original glory. In the days well before Galileo and Newton, science and religion were not separated. They worked together to discover the truth and while the latter had God as its final destination, the former had God as its starting point. Science is based on the irrational (unproven) belief that the world is intelligible along many other assumptions. This poses a serious limitation to science that can only be overcome if we accept the irrationality of the cosmos. The motto “Credo quia absurdum” holds more truth than one can ever realize at first glance. There is nothing logical in logic, whereas there is deep wisdom in the irrational. For while the former tries to build castles on moving sand, the latter digs deep inside the depths of existence itself in order to build on the most concrete foundations that there can be: the cosmos itself. The only way forward is backwards. Backwards to a time when religion led the quest for knowledge by accepting what we cannot know, rather than trying to comprehend what we do not. Science was anyway based on that in the first place.
Η ιδέα ότι η επιστήμη είναι η καλύτερη ή η μόνη οδός για να φτάσουμε στην αλήθεια για τον κόσμο μας, είναι μια βασική πίστη του μοντέρνου πολιτισμού. Και όμως, η επιστήμη έχει μεγαλώσει πολύ στηριζόμενη όμως σε πήλινα εύθραυστα πόδια. Κάθε επιστημονική θεωρία χρησιμοποιεί αξιώματα και παραδοχές οι οποίες εξ’ ορισμού δεν αποδεικνύονται. Αυτό θέτει σοβαρούς περιορισμούς στο τι μπορεί να κάνει η επιστήμη αναφορικά με την αναζήτηση της αλήθειας. Ο μόνος τρόπος να βρούμε την τελευταία είναι να αναθεωρήσουμε την πρώτη στην πρότερη της δόξα. Στις ημέρες πολύ πριν το Γαλιλαίο και τον Νεύτωνα, η επιστήμη και η θρησκεία δεν ήταν διαχωρισμένες. Εργαζόντουσαν μαζί για να ανακαλύψουν την αλήθεια και ενώ η δεύτερη είχε το Θεό ως τον τελικό προορισμό της, η πρώτη Τον είχε ως αφετηρία. Η επιστήμη βασίζεται στην (παράλογη) πίστη ότι ο κόσμος είναι κατανοήσιμος, μεταξύ άλλων παραδοχών. Αυτό θέτει ένα σοβαρό περιορισμό στην επιστήμη που μπορεί να ξεπεραστεί μόνο εάν αποδεχτούμε τον παράλογο του Κόσμου. Η φράση “Credo quia absurdum” κρατά περισσότερη αλήθεια από ό, τι μπορεί κανείς να συνειδητοποιήσει ποτέ με την πρώτη ματιά. Δεν υπάρχει τίποτα λογικό στη λογική, ενώ υπάρχει βαθιά σοφία στο παράλογο. Γιατί ενώ η πρώτη προσπαθεί να χτίσει κάστρα σε κινούμενη άμμο, το δεύτερο σκάβει βαθιά μέσα στα βάθη της ίδιας της ύπαρξης, προκειμένου να στηριχτεί στα πιο στέρεα θεμέλια που μπορεί να υπάρχουν: τον ίδιο τον Κόσμο. Ο μόνος δρόμος προς τα εμπρός είναι προς τα πίσω. Πίσω σε μια εποχή που η θρησκεία οδήγησε την αναζήτηση της γνώσης αποδεχόμενη αυτό που δεν μπορούμε να ξέρουμε, αντί να προσπαθούμε να κατανοήσουμε αυτό που δεν γνωρίζουμε. Η επιστήμη ούτως ή άλλως βασιζόταν σε αυτό εξ’ αρχής.
1. INTRODUCTION
Science today is thought to be the cornerstone of human civilization. People trust science to find solutions to their problems, to search for explanations on how the cosmos is working, even to reach the ultimate goal that was the holy grail of human philosophy for thousands of years: Truth. However, science is inherently limited and cannot play such a central role in the search for answers. This paper will present arguments to justify this proposition and will also try to explain that the untrustworthiness of science can only be supplemented by religion in its most pure form. As Einstein once postulated, “Science without religion is lame, religion without science is blind”. The pages that follow will try to extrapolate on that line of thought and delve into the depths of the foundations of science, which surprisingly contain more religiosity than one would expect.
2. PURPOSE OF THE STUDY
The purpose of this study is to show that the foundations of modern science are not capable to support a method of thinking which can positively answer any of the great metaphysical questions of humanity; questions related to the “why” and not the “how” which is usually the subject of modern natural sciences. In order for us to reach an understanding of the cosmos, we need something beyond an axiomatic-based science. This paper will show how and why such a way of thinking can only be found within the realm of religion.
3. RESEARCH METHODS
The topic under analysis was examined with the help of three tools: Scientific analysis, Philosophy and History. The latter was the tool that provided evidence for how science has progressed into what it is today. That analysis provided insights on what are the main limitations of science due to the axioms it is using. These axioms were further analyzed to show what Gödel has showed many years ago: That an axiomatic-based theory can never be proved consistent or full. The analysis was performed based on the basic premise that “If something needs an assumption to be held as true, then it cannot be considered as true per se” (this should not be considered an assumption but rather a tautology: what is not proved to be true cannot be considered true). Finally yet most importantly, the research tried to find the prerequisites for a method of thinking that could overcome those obstacles and regain the trustworthiness of humanity to science as a way to reach the truth. The path revealed is described at the last chapters of the current thesis.
4. FINDINGS
The present research analyzes the foundations of science and shows that they are unreliable markers for the quest for truth. This is done by examining not only one of the major exact sciences of today (physics) but also the major tool utilized by all exact sciences to elaborate on the workings of the cosmos (logic). The results of the exercise portray a picture that is widely different than the one promoted by mainstream science today: The holy grail of modern science (the “truth”) cannot be found unless we stop searching for it…
In summary, the scientific method [1] tries to examine which assumptions best fit the observed phenomena[1] and then formulate the best scientific model that could explain those phenomena in a self-consistent way. The method (e.g. logical deduction) or the tools (e.g. statistical analysis) used to perform the modeling of the cosmos are not of importance here. What is important is that for the scientific models to exist, there need to exist some assumptions which scientists take for granted. Nothing can be built based on nothing (except ‘Nothing’ of course, but Parmenides would have objections on whether this exists).
What will be shown is that these foundations of science (assumptions, a.k.a. axioms or principles) do not have any inherent truthfulness or validity. And if the foundations are not to be trusted to lead us to the truth, then neither science as we know it today can be trusted as well for that task.
4.1 Axioms: From Euclid to quantum physics
The Elements of Euclid constituted the first time a sector of mathematics[2] was founded on axioms. Euclid made a great work in formulating a specific set of principles on which his theory was based upon and this way of thinking dominated the western scientific thought for millennia. Even today, every scientific field from mathematics to physics and chemistry is based upon specific axioms and scientists struggle to create (or discover – depending on the philosophical theoresis you adhere to) new theorems based on those axioms.
Terminology note No. 1: Axioms are also known as ‘principles’ in other fields of science besides mathematics. It is important to note that the name really makes no difference. What is important is that we speak about unproven assumptions that we take for granted when developing a theory.
It is important to note that ‘unproven’ does not mean that there is no evidence whatsoever for the axioms or principles used. It simply means that these axioms are not proved with the certainty one would want to declare them ‘true’ in the philosophical sense of the word, meaning valid in every possible case. Nobody starts pulling axioms out of thin air because he just thought about them; usually there is a long process of thought involving both observation and induction reasoning to derive a specific axiom. (Later on, we will see how indeed this could be futile anyway, but for now let’s assume that this process is of great significance) So for example, physics contains the principle of conservation of energy, which may be proven in specific cases but which has certainly not been proven for every possible case. This is clearly self-evident, since only a specific set of observations has been conducted so far. Drawing a general universal principle based on such a limited set of information (limited in relation to what we will discover one million years from now and surely limited in relation to the extent of the cosmos) is inductive in nature and can in no case substantiate the principle as a ‘fact’ beyond any doubt.
In other words, the fact that axioms are based on scientific observations does not mean that they are proved, as a theorem is proved in geometry for example. It is true that most axioms or principles are evidence-based, but this means nothing to a scientist – proof of something entails a long and painful process that would make certain that something holds true in every possible scenario. Even though the conservation of energy is something that seems to be true based on what we have seen so far in the physical systems we have examined, we cannot possibly be absolutely (100%) certain that this principle is true in the whole universe or at every possible circumstance. Our “facts” are based – at best – on circumstantial evidence; examining all the possible physical systems in the whole universe is something we will never accomplish anyway. Hence, the conservation of energy is a ‘principle’ and not a ‘fact’.
Terminology note No. 2: Principles of physics (and other areas of science) are many times called ‘laws’. This can be really confusing to the uneducated reader, since the word ‘law’ implies something that has been proven to exist in the whole cosmos (universal laws). One should always keep in mind that regardless of the name used, we are still speaking about unproven assumptions that we take for granted when developing a theory.
Nevertheless, axioms or principles are a basic and fundamental element for science. They offer the foundations on which theories are built and evolved. Without these foundations observations are just set of meaningless data. Axioms offer the context in which a theory is built; the basic building blocks on which the whole structure of the theory is erected upon. Change those axioms and you end up with a different theory based on exactly the same observational data. It may be the case that some new evidence disprove one axiom/ principle, but in that case another axiom takes its place for the whole structure of theories to still be able to stand tall. Needless to say that this is neither a bad nor a good thing. It is just how science works. Any journey needs a starting point and these axioms offer that point of reference from which one can watch the cosmos and derive meaning from it.
This section will first examine the use of such axioms and principles in various cases of scientific endeavor. I will then show that such axioms cannot be reliable foundations for the philosophical quest for the truth.
Important Notes
The term ‘axioms’ and ‘principles’ will be used interchangeably in this paper depending on the context. In essence, even though the former are used in mathematics and the latter in exact sciences, they denote the same thing: unproven assumptions that we use as a starting point to build a theory.
4.1.1 The case of the 5th axiom
The fifth axiom of Euclid (also known as the “parallels’’ postulate”) was accepted for centuries as self-evident [2]. Why shouldn’t it be anyway? It is self-evident (this is a key phrase used for all axioms) that parallel lines never meet. And it is also self-evident that from one line only one parallel line can be drawn. It is quite interesting though that people – from the very beginning – felt quite uneasy with that axiom (and not the other four). Somehow, they felt (intuition?) that something was wrong with this axiom. That is why there were many attempts to deduce it from the other four as a theorem, with all the efforts failing. So the years passed and for centuries after centuries this axiom was part of the geometry everyone knew and used.
What is important in the case of the Euclidian geometry, is that geometry was not just ‘geometry’ back then. It was the tool used for doing math as well. Long before algebraic notation, mathematicians used geometry to calculate the squares and cubes of number simply by designing… squares and cubes. In that sense the Euclidian geometry and its validity was often correlated with the axiomatic validity of the known mathematics. This idealistic picture was soon to change with changes not only in mathematics, but in geometry as well.
Eventually, it was proved that the 5th axiom cannot be proved, i.e. that it was indeed an axiom and not a theorem based on the previous axioms. This opened the path towards other types of geometries in which we either have more than one parallel lines to a certain straight line, or no parallel line at all. Non-Euclidian geometries soon gained momentum and many practical applications to various sections of science, like in the Theory of Relativity in physics, were revealed. Again, the details of the other geometries are mute. What is of essence here is that the change of an axiom is not a matter of change in the observable phenomena, but purely a matter of choice. Can you imagine of a geometry where there are only three parallel lines to a straight line? There you go! Congratulations on your new geometry!
Axioms turning into dogmas.
What does the above example show us? In short, the obvious: The more one thinks of something as self-evident the harder it is for him to discard it. It took us literally thousand years to change the 5th axiom of Euclid and yet, the moment we did, the new way of thinking bred many new children. What was initially the basis of scientific thinking, ended up hindering scientific thinking. This is expected and totally human in nature. Our mind is very prone to prejudice. One needs to actively pursue being open-minded in order to be so.
How can we change axioms though? This is a question that could baffle people outside the domain of science and yet it is one of the easiest questions to answer. Quite simply, an axiom can change if you just change it. There is no other way to select a different starting point for your theories besides selecting a different starting point for your theories. It sounds tautological in nature and it is; tautology is anyway the only truth we can be certain about from a philosophical point of view. This way of changing an axiom was used when the 5th axiom of Euclid dead-end was reached: In an extreme simplification of the story, people simply decided to see what would happen if they opted for a different thesis regarding the existence of parallel lines and new geometries emerged based on these new options. The new axioms were as unproven as the old one. And yet, they too helped build a new theory which seemed consistent (waiting for Gödel to prove otherwise – see below) and had practical applications.
The open questions regarding what is valid and what is not valid were just made more intense with the new ‘discoveries’. If it is so easy to change a basic axiom and produce equally valid theories, then what does that imply for the truthfulness of the axioms per se? Are they just ad hoc selected cornerstones for the theories we want to build and which can be changed with others at will? Are we limited in any way regarding their selection? If such a change can easily happen in one of the most important and fundamental areas of science what does this mean for other areas where the basic principles are still in open debate? One could argue that other areas of science – like physics – are based on evidence; but again wasn’t the parallel lines axiom based on purely empirical evidence that two parallel lines never meet? And let us not forget the practical applications of the (now) three different geometries we have, indicating a strong connection of the theory to actual life which cannot be easily ignored.
Having in mind all of the above, it is important to examine the other basic cornerstone of science besides mathematics: Logic.
We must forget how to think in order to think!
~ Harmonia Philosophica
4.1.2 Logic as Organon
Aristotle founded Logic as an Organon (Όργανον, Greek for ‘tool’) in his work with the matching title (“The Organon”) for the first time [3]. In this work he managed to document the basic types of logical syllogisms and formulate the basic rules with which these syllogisms could produce valid conclusions based on some initial premises. This work helped formulate the basis of mathematical logic for the next centuries, only to be amended with new elements rather recently. Even today when we think logically we instinctively use these rules, which are generally so much accepted and embedded in our thinking that are almost automatically seen as valid. However, as in the geometry axioms mentioned above, extra caution should be exercised whenever something is seen as self-evident.
Despite the extensive use of logic from the time of ancient Greece until today, the actual usefulness of logic eludes us. Even Aristotle himself did not know how this ‘organon’ would be used for. For some it would be seen as a linguistic instrument to drive conclusions without any relevance to the philosophical concept of the “truth” whatsoever; a merely game which allows us to create new conclusions based on the (potentially erroneous) input we give to its mechanism. Anyway, it is true that logic is as good as its premises. For others, logic is the basic tool to reach the truth and the high peak of human thought throughout our history on this planet. To understand what is the case we need to take a step back and see logic from a different perspective.
If we examine logic more carefully, we will discover that even logic has axioms on its own. For example, the Law of Excluded Middle (LEM) is one of the basic laws used not only by Aristotle but also today by modern mathematics: In essence this law (remember, people have the bad habit to refer to axioms/ principles as “laws”) states that a logical proposition is either True or False (but never both or neither). This sounds like very logical and common sense, however in science nothing is (or should be) common sense. That is why this principle is used as an axiom in logic; it was also the basic element of Peano arithmetic, which is what we essentially use today in mathematics. Brouwer however chose not to accept this as self-evident in his intuitionistic mathematics and in this way attempted to build a different area of mathematics, even though historically this did not work out as intended due to mainly non-mathematical reasons.
4.1.3 Axioms in modern mathematics
As mentioned above, modern mathematics are based on axioms; with the more important ones being related to mathematical logic. Those axioms played (and still play) a major role in the attempts to formulate the foundations of mathematics in logic, an attempt the beginnings of which many trace back to Russel (Logicism). These efforts – regardless (or perhaps because) of their inconclusiveness – were very fruitful in seeding the scientific thought with perspective regarding what is or what could qualify as a founding principle for a specific field of knowledge.
This debate over the foundations included many other principles of mathematics. For example regarding set theory – initially proposed by Cantor and regarded as one of the building blocks of modern arithmetics – there were (and still are) many discussions regarding the overall validity of the set theory and its premises. Cantor started his quest for the theory by doing what every scientist should to: Question the obvious. And the obvious in this case stated that the whole bigger than the parts. This simple and ‘self-evident’ truth was at the end discarded; we now ‘know’ that the set of e.g. integer numbers is not larger than the set of positive numbers which are just a part of the first set. The counter-intuitive basis of Cantor’s theory caused great dispute, with the most infamous one being the dispute between Kronecker and Cantor. The results of their clash resulted in the modern set theory being accepted across the mathematics society, something we today take for granted. It is interesting to note here that foundations forming with a bang usually end up being accepted in silence by the generations to come.
Questioning the obvious is hard in the context of this silence. After the foundations of algebra took shape, modern mathematics based on the set theory were also formulated. But the Set theory – as any other – has axioms. Axioms leading into paradoxes (see Russel’s paradox). In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell’s paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for “choice”, and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded [4].
There are many equivalent formulations of the ZFC axioms; for a discussion of this see Fraenkel, Bar-Hillel & Lévy 1973. For illustration purposes only, a set of ZFC axioms is documented below. The following particular axiom set is from Kunen (1980). All formulations of ZFC imply that at least one set exists. Kunen includes an axiom that directly asserts the existence of a set, in addition to the axioms given below (although he notes that he does so only “for emphasis”) [4].
ZFC axioms list [4]
Axiom of extensionality
Axiom of regularity (also called the axiom of foundation)
Axiom schema of specification (also called the axiom schema of separation or of restricted comprehension)
Axiom of pairing
Axiom of union
Axiom schema of replacement
Axiom of infinity
Axiom of power set
Well-ordering theorem (the Axiom of Choice is equivalent to that)
One can easily find out the axioms of any other theory; they are indeed a fundamental element for every theory anyway. As mentioned already, their existence per se implies nothing regarding the validity of the theories built upon them. All theories which have their foundations on such unproven theses can and do have practical applications.
After someone realizes the plethora of axioms used by a theory, some ‘simple’ and ‘obvious’ questions come to mind: Can there exist a theory that is based on arbitrary axioms and not have such practical applications? Should we choose the axioms with a specific ‘common sense’ (or instinct) so as to have ‘valid’ theories? Is there a limitation on how we choose axioms or is it that ‘everything goes’, to paraphrase Feyerabend when he was talking about the anarchy which rules the scientific method?
By examining other realms of science and, especially, evidence-based ones will allow us to easier detect the methodological limitations (or even worse, the inexistence of such) of modern science.
4.1.4 Axioms in modern physics
For many, mathematics is simply a tool and not a science per se, at least not as physics is. The first can be based on arbitrarily selected axioms but the latter is based on observations and evidence. This picture is totally false. By examining how physics formulates and adheres to specific principles it will be made evident that physics also has its own axioms that can be changed much more easily than many people believe. To exhibit that, we first need to show the way physics selects these principles and show how potential changes in those principles can indeed lead to equally valid pictures for the cosmos, in a similar way as the change of axioms in geometry resulted in equally valid geometries.
To begin with, physics is – as geometry – based on several assumptions. These assumptions, when used widely are promoted to what we call ‘principles’, which play the similar role that axioms play in mathematics. Some of these principles are difficult to detect since most of them (especially the fundamental ones) are used in every piece of research without any explicit reference to it. These principles usually reflect specific philosophical dogmatic beliefs, which for many – especially those who adhere to them – seem self-evident.
As a side note, it must be noted that the principles on which modern science is based are often called ‘presuppositions’, like in the PEL (Presuppositions, Evidence, Logic) model [5]. The nomenclature though does not have any effect on the essence of the matter in hand. These presuppositions/ principles/ axioms are all some unproved statements which are taken for granted and on which we build the theories we build. From here on this paper will refer to the presuppositions as either axioms, assumptions or principles, recognizing that the word ‘axiom’ is mostly related to mathematics and geometry. The reader should be aware that what we are trying to do here is to examine the essence of the connection between science and the truth. In that quest, words are a hurdle that we must sooner rather than later overcome.
Terminology note No. 3: Principles, axioms, assumptions or presuppositions are used interchangeably in this paper from hereon – all implying unproven assumptions that we take for granted when developing a scientific theory.
Assumptions that underlie any science – and physics in particular – may include:
A world external to mind exists;
This external world is not chaotic, it is organized;
The external world is knowable (intelligible), including the world beyond appearances; and, last but not least,
(iv) This external organized world is only material. [6]
These principles need to be accepted in physics in order for any scientific endeavor to have meaning. One cannot start measuring things in a lab if he does not believe that all things are measurable. There is no reason for the scientist to start analyzing the cosmos if he does not believe that there is some kind of order in what he is analyzing. There is no meaning in doing physics (and science in general) if you do not believe that your limited view of the cosmos has any validity whatsoever. There is no point in analyzing matter to discover how the universe works, if you do not believe that the universe is created by matter and matter alone.
If these seem very similar to specific philosophical ideologies, you would be right.
Philosophy is in any case the mother of science.
And the latter, as any spoiled child, want’s to ignore (or kill, if possible) its parents and behave as if it has done everything on its own. However it is this oblivion of the foundations of science which makes it so arrogant as to believe that it is founded on nothing more than ‘facts’; a current worldview which gives birth to the monster of modern scientism. Taking the abovementioned principles one by one, one can see that the basis of science is faith. Not faith in a supreme being, but in everything that makes the existence of such a being necessary: The existence of eternal universal laws, a universe in order and the ability of tiny humans to understand and comprehend the above. Newton did believe that he could understand the cosmos and he blatantly admitted that what he did was nothing more than reading the mind of God himself.
Planck did mention that if religion aims at finding God, science starts from the belief that God exists. The principles selected are selected based on nothing more than philosophy and saying otherwise would just blatantly ignore the elephant in the room: There is no way to ‘prove’ that the universe is comprehensible (intelligible) and that our limited view (via our limited senses) is correct or not. There is no way to ‘prove’ or consider as ‘self-evident’ that us ‘unimportant’ (based on the modern materialistic view) humans can in any way confer any truth from what we see with our limited senses and limited brains. Science accepts that our senses work properly. Every aspect of science accepts that the cosmos is intelligible [7]. All scientific work is based on our belief in the intelligibility of the cosmos [8] and (this goes without saying) our ability to comprehend the cosmos [9]. If we didn’t believe in that intelligibility, there would be no starting point at all to collect evidence on which science is based upon. If we did not believe that the cosmos is intelligible there would be no sense in trying to understand it anyway. And yet, for philosophers, the very notion of our ability to sense the cosmos is under scrutiny with no specific conclusion drawn yet. Such principles make as a whole the idea of scientific realism [5] which is currently the ‘dark matter’ of modern exact sciences.
The principle that all things we need to know are measurable is yet another axiom (principle) on which modern science (especially exact sciences) is almost explicitly based upon. But who can verify that what we need to know is indeed measurable? This idea is strongly related to the notion of materialism which transcends all modern science. Still, many – if not most – physicists today work in experiments while taking this principle as obvious. Another principle strongly related to the previous one is the mechanistic view of the cosmos. Even though it is centuries old and obviously not in any way proven, all scientists today look at the cosmos as if it was a well-designed machine the inner workings of which we can understand by analyzing its parts. And let’s not even go to more fundamental principles for which philosophers still debate: For example are our senses a valid window to the cosmos? Do they work in a valid manner that could lead to our truthful understanding of the universe? Or are they a distraction from the truth of the cosmos, a distraction which makes us focus on phenomena rather than the essence of existence? The catalogue could go on forever.
And there are many more principles to add to the above. The fact that science today does not mention them explicitly makes it extremely hard to detect them, but they are there. Poisoning our thought to the point that we believe that it is not poisoned. Take for example the simple notion of ‘Things change’. If Parmenides was alive he would argue that this is an assumption we should not take lightly[3]. But science does believe in the notion of ‘change’. If it did not there would again be no meaning in doing science. (Religion on the other hand is very close to the eternal view of One made by Parmenides – this will be examined further in the latest chapter) The existence of time is another basic axiom (principle) used by modern physics. Even though time is one of the most elusive philosophical notions stirring wide debates among philosophers on its existence, physics takes it for granted and attempts of scientists to disprove its validity (e.g. Gödel who eliminated time in one of his proof for General Relativity in a revolving universe) are just considered as interesting and yet meaningless games.
The basic idea that all the above try to illustrate should be evident to the reader at this point. Non-proven philosophical principles are currently driving the quest for knowledge and we should be very careful as to how we use those principles.
Humans should try to think freely. But is this possible? Can we ever think without any axioms? Can we formulate science without principles? The thesis of this paper is that we can by using science. Although this science was not actually called that way a long time ago…
4.2 Questioning the assumptions
How axioms/ principles change in order to give birth to a new theory is a matter of debate. Some believe that the changes can only happen within a very limited frame set by our experiences and the observational data at hand. Extreme voices speak about our ability to arbitrary select axioms. The truth – despite to what one might expect – lies not in the middle.
4.2.1 Which assumptions? There are no assumptions!
As already mentioned, axioms are not provable principles which we use so, by definition, they cannot be not related to what we call truth. The notion of truth for philosophy has been referring to eternal certainties and one can never be certain if the axioms he uses are related to anything true. It might be the case that in a specific case an axiom is accidentally related to the truth, but this would only be accidental. There is no way to systematically and scientifically know whether a specific axiom/ principle holds any special connection to the notion of truth whatsoever. And what is important is that the non-existence of this relationship does not affect at all the use of the theories built on those principles.
Some examples will better illustrate this.
4.2.2 Questioning the obvious
Science today is based – as mentioned already – in many assumptions. These assumptions (presuppositions) are so fundamental that rarely do people question them. Isn’t it obvious that you see a table? Isn’t it obvious that crossing the street without looking for incoming cars might get you killed? Do you need more evidence or proof to say that “this chair that you see with your own eyes” exists? Discussions about such things usually end up in arguments using the ‘obvious’ as evidence of the obvious. Science is based on some very few presuppositions which in any case are closely related to what we call ‘common sense’ [5]. Can or should we question common sense? Even children have it [5].
But the obvious can be a trap for free thinking.
Are we to trust a kid’s opinion on the cosmos because it is pure? Fine. But let us also trust its opinion that unicorns exist (by the way, as Parmenides said, we wouldn’t be able to even talk about them if they did not exist). Are we to trust common sense on the evidence we have and their interpretation? Should we trust this common sense when it cries out that someone must have created this cosmos? Are we to trust our sensory input? But what happens if someone else has a different sensory input? Are we just going to discard people with different common logic as ‘crazy’ so as to verify our own perception? Can we say that our senses reveal anything about the cosmos when they are so limited? Touch a table. Feel it. And yet, the table is consisted of atoms which are essentially empty. What does that say about the validity of the sensory input that we get? Touch that table again. Do you sense the bacteria crawling on it? If not, what does this mean about the validity of this sensory input? Are we to trust our belief into an objective cosmos and laugh at those who claim that this cosmos is just an illusion? If we have no solid proof, there is no room for laughter. What is common sense today is history tomorrow. Quantum mechanics has provided many counter-intuitive examples which totally break down what we would call ‘common sense’. And no, we cannot use the argument ‘if we question our senses then we do not have science’ as an argument. We cannot have the result (science) as an argument to justify itself.
Let us continue the analysis by questioning some more obvious premises of science.
4.2.3 Are there axioms of higher validity?
As already mentioned, axioms do not hold any special connection to the notion of truth. Having said that, one would wander what could be the limitations in selecting those axioms be. And if he is unable to find such, he would be right: There are no limitations. Any set of whatever axioms or principles can help us build a new theory (by logically deriving theorems and theories from those axioms). However, can arbitrarily selected axioms lead to anything of use? Could randomly selected axioms lead to the formulation of a useful theory?
Even though axioms and principles are seen as non-provable propositions, nevertheless there has always been a common belief among scientists that they must have some connection to nature and reality. (by the way, the belief in the existence of an objective reality is another major assumption of modern science) For example, when recently three physicists discovered a way to connect eigenvectors and eigenvalues [10], thus contributing in a major way in the field of mathematics, this was hailed as another proof that mathematics must in any case be connected with nature. The same applies for physics, which is much more interrelated to nature than mathematics.
To examine the validity of the proposition “All axioms must have a connection with nature to produce useful theories” (where by nature we mean the connection with reality) we must find some counter-examples. If we do find cases of theories which do not have any connection whatsoever to nature and yet produce useful theories, then we would have shown that having a connection with nature is not a required component for a practically useful theory.
Such examples were already analyzed in the previous section and we can easily find more. For example the mathematics we know have multiple practical applications and they contain negative numbers, which we learn to handle from first grade. The mathematics we know have multiple practical applications and contain irrational numbers, which we learn to handle from fifth grade. And yet, there is nowhere in nature that one can find -1 oranges…
Besides the above, it is true that if we follow the modern scientific paradigm on the value of random processes in producing valid results (e.g. random mutations play a vital role in the creation of life, at least according to the main paradigm in biology, always through the rules of natural selection) or on the possibility of creating the whole cosmos from random fluctuations (see for example the theories which want the cosmos to be created from random quantum fluctuations of the void of space), we can easily think of random axioms being the seed to create something really powerful and magnificent. If random fluctuations can produce a cosmos with universal laws science can discover, why deny the possibility that randomly selected axioms can create a useful theory? (something which in any case is much less difficult to exist than a cosmos so finely tuned and with so universal laws as our own)
To this, we must add another important note: The notion of ‘usefulness’ is something that is subjective. Even the most useless of theories for me could be extremely useful for someone else and vice versa. Surely today’s materialistic civilization finds modern science theories as useful, but again a more spiritual civilization could render all those theories as completely and utterly useless. Having accurate GPS signals does not solve human everyday problems anyway.
Last but not least, why in any case should we use the criterion of the ‘usefulness’ in the first place? (remember, questioning everything also entails questioning yourself) The most critical and important things in life are useless. Or at least we do not currently know their usefulness. Are our grandparents useful? Are we useful to anyone else in the cosmos? Is life per se useful? Why should an axiom be judged scientifically based on its usefulness? Would that be a scientific argument to begin with?
4.2.4 Research and intuition
For Husserl, it is clear that the ultimate principles must be epistemological principles that are a priori eidetic laws (Husserl 1984, 235–36). Husserl used a rather unambiguous terminology in order to point out that there is one principle that is more fundamental than all the others. This is his principle of all principles. As Husserl said, “No conceivable theory can make us err with respect to the principle of all principles: that every originary presentive intuition is a legitimizing source of cognition, that everything originarily (so to speak, in its “personal” actuality) offered to us in “intuition” is to be accepted simply as what it is presented as being, but also only within the limits in which it is presented there”. In other words: Every originary presentive intuition is a source of immediate justification [11]. This is very close to what was briefly mentioned in the beginning regarding the epistemological anarchy proposed by Feyerabend. And was also the cornerstone of the vision of Brouwer regarding the foundations of mathematics.
Poincare was also a proponent of the use of intuition in science [12]. The pre-intuitionists (in which Poincare was included, others use the term semi-intuitionist [12] [13]), as defined by Luitzen Egbertus Jan Brouwer, differed from the formalist standpoint in several ways, particularly in regard to the introduction of natural numbers, or how the natural numbers are defined/ denoted. For Poincaré, the definition of a mathematical entity is the construction of the entity itself and not an expression of an underlying essence or existence. This is to say that no mathematical object exists without human construction of it, both in mind and language [14].
This sense of definition allowed Poincaré to argue with Bertrand Russell over Giuseppe Peano’s axiomatic theory of natural numbers. For example, there had been discussions on whether the property of induction is valid or not.
The five axioms of Peano are [15]:
Zero is a natural number.
Every natural number has a successor in the natural numbers.
Zero is not the successor of any natural number.
If the successor of two natural numbers is the same, then the two original numbers are the same.
If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers.
The fifth axiom is known as the principle of induction because it can be used to establish properties for an infinite number of cases without having to give an infinite number of proofs. In particular, Peano’s fifth axiom states [15] [14]:
Allow that; zero has a property P (= it belongs to natural numbers);
If every natural number less than a number x has the property P then x also has the property P.
Therefore every natural number has the property P.
This is the principle of complete induction, which establishes the property of induction as necessary to the system. Since Peano’s axiom is as infinite as the natural numbers, it is difficult to prove that the property of P does belong to any x and also x + 1. What one can do is say that, if after some number n of trials that show a property P conserved in x and x + 1, then we may infer that it will still hold to be true after n + 1 trials. But this is itself induction. Hence, the argument is a vicious circle. From this Poincaré argues that if we fail to establish the consistency of Peano’s axioms for natural numbers without falling into circularity, then the principle of complete induction is not provable by general logic. Thus arithmetic and mathematics in general is not analytic but synthetic. Logicism thus rebuked and Intuitionism is held up [14].
To cut the long story short, the issues debated by intuitionists and constructivists in relation to the validity of long held assumptions in mathematics are well-known [16]. The discussions surrounding this matter are extremely interesting for someone to follow; there have been for example cases where there is debate of whether using the Platonic system of numbers we use (instead of the intuitionistic one) results in our difficulty in describing the notion of time in physics [17]. The purpose of this paper is not to describe them all, but just to document the very existence of such debate even for many self-evident and widely used principles.
No principles can be accepted without faith in their validity (again, this is not an argument but a tautology actually). And this faith by itself is based on intuition. If we are to trust the intuition we have however, then we must come in terms with some very uncomfortable truths regarding science that could lead us – temporarily – away from it. Truths which can result from the decomposition of science as we know it, based on the various tools we currently know (like induction) and its re-composition based on a much ‘cleaner’ view of things as they are. In metaphysics, we anyway deal with things that exist ‘as they are’. And the main sector of human thought which deals with metaphysics has always been religion [18].
It could be that the correct path is not to discard science, but to rediscover it.
The next lines of this paper will discuss how this can be performed.
4.3 Irrationality as the path to follow?
The question is simple. And, thus, as all simple questions, exceptionally difficult to answer.
How can mathematics and science progress and change paradigm in order to be able to reach the truth? The answer could be more irrational than we might have thought, but because of this, much more logical than it could ever be if we chose to follow logic. Because in any case logic will have rules, which you will have to accept on the irrational basis of belief in them. Whenever you start using assumptions then the whole system you have developed will always be in question. But the truth cannot be in question. If you use axioms and principles that cannot be proved, then your theorems will never be proved as well. But can the truth be left unproven?
Modern man understands that the above lead to a different path than the one we have taken thousands of years ago. And he is just afraid to follow that path because of the uncertainty it poses. (do not underestimate the more practical reasons which have a lot of times hindered scientific progress: the fact that all science and scientists today are based on these assumptions that we now question! It is hard to get funding to research on why all current research could be obsolete…)
Questioning things is a difficult path that few can or are willing to take. It is much easier to stand behind unproven assumptions and then move on with confidence as if you have proved everything. Modern civilization’s arrogance along with deep hostility towards anything religious, have fed the monster of scientism to a level that it is no longer distinguishable from science per se.
And we must kill the beast if we are to go back to the right path…
4.3.1 Untrustworthiness of axiomatic-founded science
All justifications of science using assumptions like the ones mentioned above are essentially using the same reasoning: “These things are common sense. If we cannot trust them, what can we trust? Even kids can understand that these things are true”.
But are we really ready to go to kids for the foundations of science? Children, with their raw wisdom, can be truly terrifying for modern scientists. Because kids can be irrational. And as they can “know” that 1 + 1 = 2 (something which took more than 300 pages to Whitehead and Russell to prove by the way, again based on unproved assumptions/ axioms) they can also easily know that unicorns exist.
People today do not believe in unicorns or miracles.
Because we are too logical to know life.
But we do believe in assumptions which lead to research funding…
Because we are too vulgar to know death.
Others could also argue: “We could not even have science if we questioned such fundamental things as the input of our senses”. This especially is a very common and popular argument against any attempt to question the assumptions of science. This argument is based however on the dogma that science (as this term is referring mainly to the ‘exact sciences’ today) is and should be the only tool to search for the truth (a.k.a. ‘scientism’, which we must remember that it is a dogma and by no means a proven position).
The answer this argument is simple: So?
Why should we not accept the obvious because we could not then have science? Yes, the fact that we cannot prove the validity of our senses for sure makes us more anxious. But, so what? Should that make us not accept the elephant in the room?
It is crucial that we not lose sight of the questions science cannot ask, let alone the ones it cannot answer [19]. If we try to free our thought from axioms and dogma’s, then perhaps we should also try to remove the biggest dogma of them all: that science, as we know it today, is the only method of thinking that can lead to answers.
Searching for the “new” science that could overcome the limitations of existing science is the next logical step in our quest.
4.3.2 The vision of Brouwer revisited
In mathematics, the battle between Platonists, formalists and constructivists/ intuitionists in mathematics still rages on. Even though the formalists have officially won (there is no school today questioning the law of excluded middle or teaching that formal mathematics may not lead someone to valid proofs), specific elements of other ways of thinking still persist. We are still taught that we should construct our proofs, we are still watching in awe when a mathematician finds out a new theory or develops a new idea (see for example the Riemann hypothesis [28]) by intuition alone and others struggle for centuries to prove his thesis (see for example the Riemann hypothesis). Despite the use of irrational numbers and negative numbers every day, we still discuss about philosophy and the idea of truth and reality, regardless of the practical usefulness of such constructs.
Platonists – Constructivists – Formalists definitions: Platonists believe that the ‘truth’ is somewhere ‘out there’ for us to discover. On the other hand, constructivists believe that only what we can practically construct is worth analyzing and talking about (that is why for them the notion of infinite has no meaning – the same goes for irrational numbers). Last but not least, the formalists care for neither of the above! They only care about the rules of the game. And as long as you set the rules, then you can play… [20] [24] [26]
In Brouwer’s original intuitionism, the truth of a mathematical statement is a subjective claim [21] [25]. Accepting this should be our first step towards realization of the need for a new science. What is obvious is the source of all potential errors. The presuppositions one uses are really dangerous. You cannot easily question what you have been taught to take for granted.
However, the vision of Brouwer is limited. It reaches up to a point, but fails to reach the final destination. For us to reach our destination, we must let go of all the restraints put on us by thinking in general. As for Brouwer the Pre-Intuitionists failed to go as far as necessary in divesting mathematics from metaphysics, for they still used principium tertii exclusi (the “law of excluded middle”) [14], we must say the same for Brouwer himself.
Even though he goes a long way in discarding basic assumptions, he still uses the most basic of all assumptions: That thinking itself is something that can lead to valid conclusions! Brower and his followers afterwards tried to formulate arithmetics without the axiom of the excluded middle [22]. In the same way, we must try to formulate science in general without the axiom that scientific thinking itself is what is needed to understand the cosmos!
It is a daring attempt, but true philosophy should not be deterred in the face of difficulty. I would say that the opposite is true: Whatever seems like the hard path, this should be the path we should try to follow.
4.3.3 Credo quia absurdum! (Shestov)
Shestov, one of the greatest thinkers of modern time, spoke eloquently about the irrational. His objections to logic were all too logical to deny. “If someone requires you to adhere to his logic in order to understand his arguments, then what value can those arguments have?” he wandered [23]. This is something so obvious that we tend to forget it as we speak about science today. Every theory we have is based on axioms and yet we always fail to truly understand what this simple fact means.
What is logical for you could be illogical for me and vice versa. And if this happens, this will not be due to a caprice, but due to fundamental differences in the assumptions we make when we start analyzing the problem at hand.
For Shestov the motto “Credo quia absurdum” holds more truth than one can ever realize at first glance. There is nothing logical in logic, whereas there is deep wisdom in the irrational. For while the former tries to build castles on moving sand, the latter digs deep inside the depths of existence itself in order to build on the most concrete foundations that there can be: the cosmos itself. And in that sense, one can only believe something not because it is not irrational (i.e. it is rational) but exactly because it is irrational!
What I mean with that will be more evident later on.
For the moment, just keep the idea lingering in your head…
To go back to the problem of the scientific foundations, it seems we are in a stalemate since there is no way to determine which assumptions are the best to use. Yet, this is one of the cases where the difficulty to answer a question denotes a third option we rarely consider in today’s knowledge-addicted civilization…
4.3.4 Religion as non-axiomatic Science
When a question – like the one above – is hard to answer, one should consider the obvious. Besides the various possible answers to the question, there might be another solution to the problem: Perhaps the question itself is wrong! Science is a way to interrogate nature [5]. And this can and should be done with no constraints whatsoever. Many people claim that science is great because it is based on so few assumptions [5]. Why not make it even greater and base a new science on zero assumptions?!
At the end, the meaning and the purpose of the cosmos are not to be analyzed with science. The truth is not to be seen with our eyes or sensed with our touch. As Jung wisely pointed out, our self is the only element of the cosmos that we experience without the mediation of anything. And this is the only truth we can ever lay our hands onto. Could it be that this is the only truth worth pursuing? Only time will tell. If time exists after all. At the end, what is certain is that we must question everything in order to reach the truth – if such a notion even exists. And this questioning should consider nothing as self-evident, not even our logic or senses. This seems like a heavy toll to pay. And it is. But the destination is of such importance that this toll is relative small in exchange.
Religion is a way to view the cosmos without assumptions or presuppositions. In that sense, it is much more scientific than science as we know it today. It just entails accepting the cosmos (weirdly enough in the same way science today accepts the intelligibility of the cosmos). Pure thought can only be possible without thought. As Shestov once explained, whenever we try to understand something we destroy it in our effort to make it fit into those little boxes we have built in our brains.
Religion is about accepting what we know. It is not about analyzing and understanding, but about being part of the cosmos as it is. Every scientific theory is based on unproven axioms, thus nothing. And this ‘nothing’ is nothing more than our acceptance of the obvious: The cosmos is here for us to know. And we can know it only because we are part of God himself! Religion has said that a long time ago. We have believed in our godly nature for thousands of years. Until we started denying our self. And resulted in a cosmos void of any meaning, except the comical belief in our science. There is no logic entailed in believing. Only the acceptance of the things your heart has already seen. We seek to understand the cosmos, but to paraphrase Pascal, we wouldn’t be searching for understanding the universe if we hadn’t already known it…
The eternal mystery of the world is its comprehensibility…
The fact that it is comprehensible is a miracle!
~ Albert Einstein [27]
5. CONCLUSION
I went for dinner with some friends from work once. At some point someone mentioned that a university did some empirical research to determine whether a tree falling in a forest void of humans will actually make any sound. I do not know whether that was a true story. I do know that everyone laughed at that though. And then, they kept on eating. But I wanted to cry. We are so much accustomed to our assumptions that even at the slightest hint of them being wrong we just… laugh!
Imagine two small worms living all their lives inside the earth. Discussing about philosophy. Being so positive that their senses are telling them all they need to know about the cosmos. Laughing at the possibility that this is not the case.
Questioning the validity of science or even our very senses seems like an exaggeration. But so is the quest for truth. If we want to play ball in the big league, then we should be prepared for some heavy loses. At the end, we might lose what we cherish the most (science), only to discover that all this time we missed something much more important.
Plato once told us about a cave. And we laughed at those inside the cave. But we never understood that we should also laugh at those outside the cave. If those inside are not getting valid inputs, what makes us so certain that those outside do? Why should the shadows inside the cave be invalid but the shadows outside it valid?
We have used to see science as the only way out of the cave.
But we have forgotten that it was originally called religion…
We have accustomed to think that thinking is the only way to reach the truth.
But we have forgotten that thinking is based on non-thinking…
Yes, the cosmos we live in is irrational.
Yes, the fact that it is intelligible seems like a miracle.
If we try to analyze it, we will end up denying it.
For there is nothing which can be said to prove it.
Accept it we must!
For we are part of it.
And believe it, exactly because of its irrationality!
We can understand the mind of God.
Only because we are part of Him!
Remember?
Credo quia absurdrum!
Science is (was) based on it! (or should we say Religion?)
BIBLIOGRAPHY
Scientific method, Stanford Encyclopedia of Philosophy, retrieved from here on 2019-12-26.
Jeremy Gray, (2017), Epistemology of Geometry, Stanford Encyclopedia of Philosophy, retrieved from here on 2020-01-07.
Aristotle, Το Όργανον – Αναλυτικά ύστερα.
Zermelo–Fraenkel set theory, Wikipedia article, retrieved from here on 2019-11-11.
Hugh G. Gauch Jr, The Resources, Powers, and Limits of Science, this manuscript was received by PNAS on 15 August 2019 and declined on 25 September.
Toomela A. (2019) Science is Based on Certain Assumptions. In: The Psychology of Scientific Inquiry. SpringerBriefs in Psychology. Springer, Cham, retrieved from here on 2019-11-10.
Alan Thomas, (2018), Intelligibility all the way down: Interpreting Nagel’s Mind and Cosmos, Klesis Revue philosophique, Vol. 41, 01.08.2018, p. 1–29, retrieved from here on 2020-01-07.
Steve Fuller (2008), Dissent over Sescent, Icon Books, p. 5.
Spyridon Kakos, (2020), Religion as the foundation of Science, submitted to IFIASA for review.
Neutrinos Lead to Unexpected Discovery in Basic Math, retrieved from here on 2019-11-20.
Philipp Berghofer, 2019, Husserl’s Project of Ultimate Elucidation and the Principle of All Principles, Canadian Journal of Philosophy (2019), 1–12, doi:10.1017/can.2019.40, retrieved from here on 2019-11-13.
Poincare, Stanford Encyclopedia of Philosophy, retrieved from here on 2019-11-26.
Gerhard Heinzmann, Philippe Nabonnand. Poincaré: intuitionism, intuition, and convention. Mark van Atten, Pascal Boldini, Michel Bourdeau, Gerhard Heinzmann. One Hundred Years of Intuitionism (1907-2007), Birkhaüser, pp.163 – 177, 2008, Publications des Archives Henri-Poincaré, 978-3-7643-8652-8. ff10.1007/978-3-7643-8653-5_11ff. ffhal-01083141f, retrieved from here on 2019-11-26
Pre-intuitionism, Wikipedia article, retrieved from here on 2019-11-26.
Peano axioms, Encyclopaedia Britannica, retrieved from here on 2020-01-08.
Davis Philip J., Hersh Reuben, (1981), Η μαθηματική εμπειρία, εκδόσεις Τροχαλία.
Nicolas Gisin, (2020), Mathematical languages shape our understanding of time in physics, Nat. Phys, doi:10.1038/s41567-019-0748-5.
Spyridon Kakos, (2010), Religion and Science unification – Towards Religional Science, Harmonia Philosophica, retrieved from here on 2020-01-07.
Giuseppe Butera, (2011), Reading the Cosmos – Nature, Science, and Wisdom, Inroduction (x), retrieved from here on 2020-01-07.
P.J. Davis, R. Hersh, (1981), Η Μαθηματική Εμπειρία (The mathematical experience).
Intuitionism. (2019). Wikipedia. Retrieved from here on 2019-10-13.
Makoto Fujiwara, Constructivism and weak logical principles over arithmetic, retrieved from here on 2019-11-27.
Λεβ Σεστώφ (Lev Shestov), Στους αντίποδες του ορθολογισμού (At the opposite of rationalism), original title: Bezpotchviennost, translated from the French edition “Sur les confins de la vie. L’Apothéose du dépaysement”, Printa editions, 2005.
Rosalie Iemhoff. (2019). Intuitionism in the Philosophy of Mathematics. Stanford Encyclopedia of Philosophy. Retrieved from here on 2019-10-13.
L. E. J. Brouwer. (2019). Wikipedia. Retrieved from here on 2019-10-13.
Formalism (philosophy of mathematics). (2019). Wikipedia. Retrieved from here on 2019-10-13.
Andrew Robinson, We Just Can’t Stop Misquoting Einstein, Prime Mind, source link, 2016.
The Riemann hypothesis, Encyclopaedia Britannica, retrieved from here on 2020-01-07.
[1] The term “phenomena” is the right term and not “fact” which is widely used in science-related articles. We can never be certain of the truthfulness of what we can experience via our senses. This is an important question that is outside the scope of this paper.
[2] Geometry and mathematics were inherently connected up to very recently.
[3] According to Parmenides, change is just an illusion. For the great thinker there is no way for A to change to B, for if it does then it wouldn’t still be A. And there is no way for something which does not exist at all to come into existence or for something which exists to perish into nothingness.
A new computational-model reveals that
serotonin, one of the most widespread chemicals in the brain, can speed up
learning. (1)
Researchers have found that piano
lessons have a specific effect on kindergartners’ ability to distinguish
different pitches, which translates into an improvement in discriminating
between words. (2)
We live in a knowledge-centric world.
Wanting to add more and more into the sea of knowledge. While what we should be
doing is trying to empty it. And see that it is a bottomless sea. With every
knowledge added, a new question arising. With every mystery solved, ten more
appearing. Like the Danaides, cursed to complete a feat which is from the
beginning destined to fail.
Stop trying to understand.
There is nothing to compute.
There is nothing to discover.
Every time you look at that sea.
You are looking at yourself on the
calm surface…
Let the slightest drop fall in and the simplest and deepest of the truths will emerge. The image is distorted. There is no you. There is nothing to see. Dive deep to the bottom of the sea. And see for yourself. There is nothing there. Only the thing you bring with you… Suddenly you are standing dry. Next to a big old tree. You hold water in your hands. You are thirsty. But… look at that small pond. There is a fish in it. You put the water inside… And without knowing it… You kill yourself…
A Flat Earther will try to fly a rocket to prove his theory right (or wrong). (source) All jokes aside, we should all respect any man (or woman) who is bold enough to try and prove (or disprove) the theory he believes in. This is something we don’t see every day, not even in science.
And this gives me the opportunity to speak about something much more important: the nature of science itself.
People laugh at the Flat Earth theory.
And rightfully so.
It is an absurd theory which wants us to be living on a flat planet, instead of the sphere Earth we all know and love.
And yet, there is something really interesting to see and observe regarding the Flat Earth theory.
And this is what is true for all theories: that it is formulated in such a way that it totally aggrees with all the empirical (observational) data!
If you pay closer attention to that crazy theory you will be astounded to notice that the people defending it do not disagree with “us” (the rest) on the facts but on their interpretation! Yes, Flat Earthers do agree that we travel from China to America on a plane, yet they explain this travel as taking place on a different world than we know! (and no, we don’t travel from North to South) Yes, Flat Earthers do agree that we have pictures of Earth from space, yet they interpret them in a different way (from a different lens to be exact) than we do.
And when you tell them to go to space and see for themselves or to go to Antarctica to believe for themselves that there is no great Wall there, they simply answer that (a) they cannot, which is for most part true and that most importantly (b) we haven’t been there personally either! Meaning that our argument to ask for evidence is answered by a similar argument which claims that we personally do not have such evidence either! (By the way it is true that one cannot easily do to Antarctica, due to a very strict treaty which – surprisingly – has been signed by almost all countries on the sake of the environment)
I don’t believe we live in a flat planet.
But yet again, I don’t believe in modern atheistic dogmatic science neither…
One man one said that modern physics can prove anything given the right theory and circumstances. And nothing could be a better testament to that than the Flat Earth theory! A theory that – all jokes aside – proves the most important thing to know about science and scientific theories: that they can be complete and consistent and yet have no relation to reality whatsoever!
It is true.
From the Flat Earth to the theory of multiverses, science history is full of crazy ideas. (And to be honest, the theory of multiverses is much more crazy)
The only question to ask is: do you believe hard enough (in scientism) to give them credit?
Related articles: Search Harmonia Philosophica for “Limits of science”, “Scientific theories” or “Scientific models”.
