Saturday, September 25, 2010

The limits of science


Limits of Science


Man has to awaken to wonder — and so perhaps do peoples. Science is a way of sending him to sleep again. ~ Ludwig Wittgenstein, Tractatus Logico-Philosophicus

1. Introduction - About Science


Exact Science has been for a long time now the tool used by most humans in order to understand the physical world. The great successes of sciences like physics or astronomy in predicting events and helping the everyday life of people has established "exact" science as the ultimate tool to discover the truth. This has unfortunately led to a decline of the importance other scientific fields of human thinking have - sectors as important as philosophy, history, religion, linguistics. We tend to forget that law or history can be as scientific as quantum physics. We tend to forget that science is seeking the truth with a systematic way and that fields of science like physics or chemistry are only a small portion of it. Science is not only physics. It is also the exploration of human affairs, the attempt to explain why humans fight each other, the struggle to understand what “bad” and “good” is. Anthropology, sociology and law are fields of science in the same way cosmology is. Before someone starts dealing with science, he must pay a visit to a good philosophical dictionary so as to make sure that he understands all words with their correct meaning. I strongly recommend the Dictionary of Philosophy, of Dagobert D. Runes (1942, New York) as a basis for your initial knowledge seeking. It is unfortunate that the abovementioned misconceptions have led humans away from thinking about humans. Fortunately, many thinking people of our time have pinpointed the problem and insist on thinking “scientifically”, rather than thinking “scientifically as physicists”. Science is not only “thinking about numbers based on physical evidence” – it can also be “thinking logically” or “thinking systematically”. And even more important is the fact that really great science is about thinking “out of the box”, it is about thinking in a way that one would consider “illogical”. The greatest scientific theories were the result of irrational thinking of great minds, who went outside the “logic” of their time…

2. About exact science: A definition


Exact science is mainly based on observation and induction. You observe one phenomenon, then you observe a similar one, then another one and so on, until you reach to a conclusion about how such phenomena work. Then you formulate a theoretical model that predicts similar phenomena in the future. If the model works, then your theory is accepted by other scientists (the infamous "peer-review" model, which can lead to weird applications of auto-censorship for not widely accepted ideas [1], but this is only out of the scope of this chapter). Until a phenomenon not following your set of rules is observed...

 
Given any rule, however 'fundamental' or 'necessary' for science, there are always circumstances when it is advisable not only to ignore the rule, but to adopt its opposite. For example, there are circumstances when it is advisable to introduce, elaborate and defend ad hoc hypotheses, or hypotheses which contradict well-established and generally accepted experimental results,  [...] there is only one principle that can be defended under all circumstances and in all stages of human development  It is the principle: anything goes. ~ Paul K. Feyerabend, Against Method: Outline of an Anarchistic Theory of Knowledge (1975)

2.1          Other fields of science


As mentioned above, besides the "exact" sciences many other fields of science also exist. These other scientific fields are based on the "logical and structured way of analyzing a specific subject" and include sectors like anthropology, linguistics, law and psychology. One of the most negative impacts of the "exact" sciences' domination in modern times is the demise of these other equally scientific fields. It is a good sign that some people are beginning to realize the mistake and a regeneration of interest in these "human" sciences is observed; in this modern era of dehumanization this is what we need the most. People realizing that there are more things than having GPS devices or new high-technology clothing which play MP3 while you wear them. Understanding our past (via History), other people (via Anthropology) or our own self (via Theology) might be far more important.

2.2         Definition of Science


Science is the systematic search for truth. And in that context physics and chemistry are not the only fields of science that exist. Many people confuse science with “empiricism” (i.e. all knowledge is based on human experience only). Empiricism is a philosophical dogma and one could dare say a very outdated one. [2] And it is of great importance to understand that many scientific breakthroughs have been conducted via totally illogical and “unscientific” (at least according to the common definition of “scientific”) bursts of genius inspiration. Outbursts which many times have nothing to do with anything that can be sensed or be part of human experience; for example, any research on the notion of ‘infinite’ or research related to the existence of ‘multiple universes’ or dimensions which we cannot feel.

3. Limitations of exact science


Exact sciences like physics have many limitations, often disregarded by their "followers" (i.e. people who believe in “scientism” and think that measuring, evidence-based exact science is all that there is in the sector of human knowledge-seeking). The most important of these limitations are analyzed in this chapter.

3.1          Axioms of Exact Science turn into Dogmas


The main limitations of science are a result of the things science takes from granted - i.e. of the axioms it uses as the basis for any further research. When you use an axiom, then you start building castles on sand. You cannot prove axioms and if you base everything on them, then you guarantee that your whole theory cannot be ultimately proved. (something proved by Gödel at the end – see his Incompleteness Theorem) Needless to say that science cannot exist without axioms. This is an inherent limitation of science which is not bad per se; the problems start when we forget that we use those axioms and start thinking of them as “self-evident truths” instead of arbitrarily chosen starting points. Some of the main (potentially wrong, according to my opinion, but this is not what is important here) axioms of exact sciences are listed below.
  1. Everything is matter. Sciences like physics have excluded anything spiritual from the "equations" of the universe. Physics axiomatically says that the cosmos consists of nothing more than electrons, protons and other particles that obey some physical laws - while at the same time it denies the existence of anything 'spiritual' in nature. In that context, no wander scientists cannot find 'evidence' for God or clues for the existence of any kind of 'Purpose' in our existence. After you have axiomatically accepted that a system consists of A, B and C only, how do you expect to find D anywhere? [2] The science of biology also makes the same mistake: it uses an axiom as a dogma without ever questioning whether it is right or wrong. Modern biology takes for granted that humans are animals and that evolution rules apply to everything, without proving it. Modern science is based on the doctrine of materialism [3], also known as "physicalism" [4]. However, we must always keep in mind that materialism is an (unproven) philosophical dogma and not a proven case. [5]
  2. Natural laws are universal and apply to everything and everyone. This, even though it has never been proven yet, is the basis of modern exact sciences. It is just something scientists believe that is true, and we have founded everything on it. [6] This axiom is also what causes problems in the explanation of the basis of our human nature: "free will". If we are to accept the existence of universal physical laws, then no free will can exist. But most of us think (or even know) otherwise. Most of us feel that we "decide" what to do. Science simply cannot explain that (if this axiom applies) and to take it for granted would mean that we will need to discard a big part of our very nature.
  3. All things are measurable. This is based on a highly materialistic view of the world and has nothing to do with reality (whatever that word means). Things like moral, emotions, aesthetics and love cannot be measured, even though they are very (or the most) important for human life. Science fields like physics, mathematics and chemistry simply cannot deal with these things.
  4. We can know everything. Many scientists wrongly believe that humans will someday come to know everything, that there is no such thing as "un-knowable" area. Gödel has showed they are wrong by proving that there are truths that cannot be proved anytime by anyone based on a specific axiomatic model.
  5. Our Logic works correctly. This may sound odd, but it was one of the main axioms used to form the foundations of science. Out logic "must" work properly if we are to trust it. [6] But whose logic is “correct”? Who defines “logic” and who defines “illogic”? Is it the majority? Is it the experts? What if the experts change their mind? What if my logic says that God exists and your logic that He does not? [the chapter of Harmonia Philosophica deals more with these issues]
  6. Everything can be replicated in an experiment. Because science needs experiments to prove or disprove theories (“theoretical models” to be more exact), it cannot deal with one-off events that happen only once and cannot be reproduced. This is a simple but rather important axiom: we believe that all phenomena can be studied by experiment or observation. What happens with things that happen only once in the Universe's history? A very good example is the creation of the Universe. If it did happen only once, how can we replicate and study it?
And here we should note that the biggest scientific discoveries were made by great thinkers, despite of contrary hard experimental data of their time! De Broglie created the theory for the double nature of particles despite that no experimental proof existed then regarding his position. Everett has created his theory of multiple worlds and he has convinced with this most scientists today, despite that this theory can never be confirmed with experimental data. Einstein thought of the theory of relativity based on mind experiments and not on physical experiments. When D.C. Miller published his experimental criticism of the theory of relativity [13] Einstein said that "I did not take them [the experimental results of Miller] seriously not for a moment". [11] When he was asked where he drew his conviction from, he said "from the intuition and the general sense of the situation" (die Vernunft der Sache). [12] [14] Even Descartes (to whom we owe the modern philosophical dogma that everything in the world can me mechanically replicated) in his great work Dioptrique found the law of refraction with the use of mathematics only and NOT with experiments. [1]
 
  1. Mediocrity and Copernican principle. Many axioms are referred to as “principles”. In cosmology, the Copernican principle, named after Nicolaus Copernicus, states the Earth is not in a central, specially favored position. More recently, the principle is generalized to the simple statement that humans are not privileged observers. In this sense, it is equivalent to the mediocrity principle, with significant implications in the philosophy of science. The mediocrity principle is the notion in philosophy of science that there is nothing special about humans or the Earth. These two principles combined form the basis of our current cosmology. [14] [15] Even though the Theory of Relativity states that we can choose ANY point of reference for basis of calculations in the universe (which means that we can easily put Earth in the center of everything and form the physical laws with that planet as a reference point), even though the universe seems the same in any direction we may look and all galaxies seem to drive away from us (as if we were in the center of the universe as Hubble himself admitted, but then denied simply because that would be opposing these two "principles"! [16]) some people still maintain their "belief" (dogma?) that humans are just grains of dust in the cosmos. A belief which is strongly instilled in our way of thinking and which inhibits our progress in any field of knowledge – a speck of dust cannot argue for morality or any other higher ideal. A speck of dust just exists. A speck of dust is indeed mediocre.
  2. Dialethism is false. All science is based on the axiom that a proposition can be either true or false (i.e. that dialethism is false; this is also an axiom of mathematical logic by the way). However, there are substantial evidence towards a different "reality". Consider for example the logical proposition which states for itself: "this proposition is false". Is it false? If yes, then it is true. Is it true? If yes, then it is false. It is like me claiming that “I am now lying”. Am I telling the truth? Graham Priest has argued that the only way out of this infamous paradox is to accept a different axiom: that a logical proposition can be true and false at the same time! [2] Other great philosophers have also argued towards the point that the distinction between "right" and "wrong" could be something imposed by the limited abilities of the human mind (see Harmonia Philosophica chapter for more on that).
  3. Time exists. People in the past may have felt that time, as Newton described it and used it in his physics, was indeed something that "exists". We think time passes. But there are more and more scientific views (see Gödel and Rovelli, to start with) are in favor of the theory that time is just an illusion. If this is correct, the implications to philosophy could be incredible. And the implications to science also would be tremendous: all our ideas about motion and events (i.e. changes over the passing of time) are based on this elusive concept of time! [3] [4] [5] [6] [7] In any case, what is difficult to be defined most likely does not exist. And time, among many other notions science uses (“particle” being one of them as well, which started from the notion of a small piece of matter and has ended up to the vague notion of a wave of energy which materializes in a mystical way through mystical procedures), is surely not difficult but philosophically impossible to be defined.
  4. The notion of "change" exists: At the time before Socrates [Gr. Σωκράτης] in Greece, the idea that things "change" was a topic of discussion between philosophers and not a matter solved. How can a thing be changed without losing its identity? Perhaps things do not change eventually but only seem to change, said Parmenides [Gr. Παρμενίδης]. The cells which constitute our body as humans are all replaced several times during our lives. How do we know that we are who we think we are? Is there a "reality" beyond what we see? Finally the theory of Democritus [Gr. Δημόκριτος] and Leukippos [Gr. Λεύκιππος] (according to which things are changing) prevailed over the theory of Parmenides, and that has defined profoundly our scientific thinking ever since. Is that what is actually happening though? [22]
  5. The whole can be analyzed if we examine its parts. This is not certain by any means. If we examine hydrogen and oxygen, we will never find out any information about the wetness of water. If the universe is holographic, as many modern physicists postulate, then by examining the parts we achieve nothing. [refer to the Harmonia Philosophica chapter for the implications the non-existence of "time" and "change" could have to the main issue that troubles people from the very start of their existence: Death]
More dogmas The dogmas of science are practically endless. Rupert Sheldrake pinpointed a few more in his lecture "The Science Delusion". An indicative list of such dogmas could include…
  • Laws of nature are stable over the course of time. For now and forever, this is what we have. Except of course from the moment of the Big Bang when all those laws were created. It is like the joke which states "Give us one free miracle and we'll explain the rest". The miracle in this case is the sudden creation of all matter, energy and fixed laws of the Universe in an instant.
  • The sum of matter and energy stays the same. (that is, except from when a huge mass and energy was created out of nowhere in the Big Bang)
  • Nature's purposelessness: Nothing in the universe has a purpose whatsoever. Everything is just pawns in a cosmic dance of lifeless matter inside a cold unconscious universe.
  • Everything which makes us who we are is in the genes (another expression of materialism) Your mind is inside your head. Memories are in the brain (even though no one knows how it works and despite the evidence we have against this idea; the dogma of materialism runs deep in every section of modern science)
  • Mechanistic medicine is the only medicine that works. No alternative theories exist.
  • Occam's razor is the criterion with which we should judge every theory. But why should the simplest solution be the correct one? If the real solution is complicated, we will never find it...
  • Balance is inherently natural. But why even think about that? Keeping balance requires much effort.
  • The observed exists. But how many of us have considered that thinking about the observed precedes the observation? One of the main problems of science is that we do not have a single clue about what the 'real' reality looks like, so that we can understand how close we are to the ultimate truth with our scientific theories. Without being able to know what the goal is, it is very hard to know if you are going the correct way [1].
  • Mathematics logic axioms are another example of axioms which can affect the way we ‘see’ the world in a significant way. An indicative list of the axioms used in the infamous "objective" mathematical logic [24] include:
Zermelo–Fraenkel axioms: These are the de facto standard axioms for contemporary mathematics or set theory. They can be easily adapted to analogous theories, such as mereology.
  • Axiom of extensionality
  • Axiom of empty set
  • Axiom of pairing
  • Axiom of union
  • Axiom of infinity
  • Axiom schema of replacement
  • Axiom of power set
  • Axiom of regularity
  • Axiom schema of specification
  • Axiom of choice
Other indicative axioms of mathematical logic:
  • Von Neumann-Bernays-Gödel axioms
  • Continuum hypothesis and its generalization
  • Freiling's axiom of symmetry
  • Axiom of determinacy
  • Axiom of projective determinacy
  • Martin's axiom
  • Axiom of constructibility
  • Rank-into-rank
  • Kripke-Platek axioms
The abovementioned axioms are the basis for the limitations of science. But as mentioned above, my objection is not in the use of axioms by itself, but in our completely forgetting that we use those axioms. When we believe that our science is based on "true" arguments, then we forget the basis of our science. True science can draw strength from the continuous questioning of the underlying axioms/ principles we use; this is something we should not be avoiding but something we should actively seek! When we forget that we use axioms (and that if we use other axioms we will reach completely different conclusions) then those axioms turn into dogmas. And dogmatism, in any form, is not a good thing...

The true mystery of the world is the visible, not the invisible. ~ Oscar Wilde

3.2         Main limitations of scientific tools


Other limitations of science, beyond the ones mentioned above, come from the tools it uses. Some of them are discussed below.

 Induction

The tool of 'induction' is by itself a problem for science. Let’s say you observe a frog and see it is green. Then you observer another frog and you see that it is green also. Then another, and another - until you are convinced that all frogs are green. Then you write a theory about frogs. Everything seems quite good up to this point, quite 'real', quite 'scientific'. Until you observe a black frog... [3]

Sensory input

Science is based on our senses. The limitations of them may pose significant limitations to how we understand the world, that we may never be aware of [1]. Since we do not know if our senses work "correctly" (mainly because we do not have a benchmark as to what is the "correct" way for senses to be receiving signals from the world), we will never know how "close" we are to "Reality".

 Logic

Science uses Logic as a tool to reach to conclusions. However even Aristotle, the founder of Logic, did not know how logic could be useful: as a tool to reach the truth or as merely a tool to analyze language and its structure? Many modern philosophers, like Wittgenstein, think that human language has many limitations and that due to these limitations, one must be careful as to talk only for things he can talk. The faith in the whole structure of science is based on the faith that logic works. If the latter collapses, then science is without any justification at all. One of the greatest mathematicians of all times, Russell, actually proved that logic has great limitations in his quest to set the foundations of mathematics. The greatest logician after Aristotle, Gödel, proved that science cannot prove it can prove things! Even logic requires faith to rely on after all...

3.3         Exact sciences cannot measure


It sounds strange to claim that "exact sciences cannot be exact", but it is absolutely true. Ask a physicist to measure the length of a table in your house. He will take the necessary tool to measure length and will measure, let's say, 80 cm length. But is this true? Is this the correct answer? Just imagine another physicist comes to your house and measures the same table with a different - but more accurate - tool. He will measure the length equal to 80.095 cm. Is this correct? Another one comes to your house and with a tool of greater accuracy measures the length equal to 80.0949988988171716 cm. Could this be our final measurement? I am afraid not. Every time you use greater accuracy, you end up with a different number! Ok, one might say. I will use a SEM and measure the length at the atomic level. Cool. Go on and try it. You will attempt to do just that only to realize that at an atomic level you stumble on the uncertainty principle of Heisenberg: you cannot know where the electrons are exactly! What does this mean? Does it mean the table has no length? It must have a length, since we see it in front of our eyes. The point here is that the table has a length, but our science cannot measure it. That is not a limitation of our science today. We know that we will never be able to measure the exact length of the table due to quantum mechanics phenomena occurring. Final outcome: you do not know and will never know the exact length of your kitchen table! Weird conclusion for what we call the "exact sciences"...

 Mathematics cannot spell numbers

From the beginning of human science, mathematics is considered to be the most crystal clear, provable, safe, well documented and better-founded field of knowledge. Due to its nature, mathematics was the first field where an attempt to set the foundations of a fully justifiable/ provable theory was conducted - and failed (see Hilbert program, Russel and Gödel's incompleteness theorem). However, mathematics have problems even in more fundamental things. Try to ask a mathematician to write down the π number. He will start writing the first digits 3.14159... and then stop. You will ask him "why did you stop?" and he will answer "I stopped because π has infinite digits and we do not know all of them. We have also proved that its infinite number of digits does not follow any repetition pattern. So we cannot write all of them". But you are not satisfied... You want to know what π is. The impressive thing about the above story is that you will never get an answer and that mathematicians feel comfortable with that! Mathematics have named some numbers as "irrational" ("άρρητοι" in Greek), meaning that they cannot be expressed in writing since they have infinite digits that will never follow a pattern. One of these numbers is π. The square root of 2 is another example of irrational number. Does that explanation is enough for the thinking human? No. When mathematics claim to be the most exact and well-founded scientific field, it sounds rather "basic stuff" to ask for a mathematician to just write a number down on paper. But this is exactly what he/she cannot do! They say that we should be happy with the 1 billion digits we have calculated for π, but can we be happy when the "measurer" cannot measure?!? This is more than just a good "trick" to play with. This shows an innate inability of mathematics to use its own language so as to just speak! And if we start with that, who knows what other inabilities we have overlooked?

3.4         Exact science does not understand reality


Exact sciences are a tool to formulate theories so as to explain what we see with our senses. Those scientific fields tell us nothing about what we call "reality". Reality is what "exists". We filter that reality via our eyes and ears and we then attempt to understand what that "world of our senses" is. Science provides models to describe physical phenomena – nothing more; nothing less. To engage into talking about reality based on science is simply a categorical mistake. (on the other hand, religion accepts the reality of what we experience raw without any need to put it into the little boxes scientists build while formulating theories; that is why for example religion easily accepts the notion of free will while at the same time science has a really hard time even thinking about it in a universe full of equations regarding lifeless particles as part of the Standard Model)

For example, things fall on the Earth. That is the reality. We sense that reality with our eyes and "see" apples fall onto the soil. We then try to explain what we see by formulating the theory of gravity. Many people think that since apples fall due to the theory of gravity and that since we see the apples, then gravity "exists" as the theory tells us. That is not correct: when the theory of gravity is proven wrong and replaced by another theory (that is happening all the time with all scientific theories), apples will continue to fall! Our theory, our interpretation of what we sense, will have changed, but that would have no effect whatsoever to the thing we call "reality". Another example is the invention of the transistor. The transistor works as the modern theories of physics say. However, when all these theories are replaced by totally new ones, the transistor will continue to work... The fact that reality continues to "work" has absolutely nothing to do with the fact that our models work or not. The former does not grant "validity" to the latter. At the point where science attempts to move from observation to general theoretical models, it enters the realm of uncertainty as Kastoriadis (Gr. Καστοριάδης) says.

4. Failures of exact science


Science like physics and chemistry have failed to give explanations for many things we see in our everyday lives. Some of them include important aspects of the reality we experience and which we consider as ‘obvious’. For example, no scientific theory exists that requires causality (i.e. that everything happening has a prior cause). As far as scientists are concerned, there could be things happening and their cause happening before them. No scientific theory exists that requires 'time flowing forward' as we feel it happening. As far as scientists are concerned, the 'arrow of time' may as well be heading backwards. Some scientists today have attempted to explain that some scientific theories (like thermodynamics) really demand the arrow of time to go forward, but not with great success. Last but not least, no scientific theory explains human goodness, human altruism. The theory of evolution - no matter how well it explains many things about species evolution - cannot explain why you may endanger your own life to rescue a complete stranger (see “Evolution and Intelligent Design - The way to an agreement” chapter). These are just indicative examples and it must be noted that we do not examine here if science is correct or not in saying specific things. For example, and as far as the second point above is concerned, time could be one of our greatest illusions and science could be right in not finding evidence to support the "passing of time" (see Gödel’s circular and timeless universe on that). However, this does not matter to the argument postulated here. What is important is the innate inability of science to give a definite yes-or-no answer to questions, an inability which is too evident to be ignored. Despite the above, it is of paramount importance to note that the fields of exact science are indeed the best tools we have to understand the physical world (at least for the “how” part). Scientists today have created really good models of nature that make predictions and can help us live better lives, work more productively, understand universe in much more detail than ever before. The criticism above is not a call to disregard science, but more of a wake-up call to realize its limitations. We must just be careful not to consider the tools of science tools perfect (or science in general as a tool perfect), while having an open mind to examine also other ways of searching for the truth. Sometimes living a better life stems out of things as simple (and thus difficult to define, measure and manage) as love and compassion, not out of having better Internet or faster cars.

5. Conclusions - Going forward


We must use exact science carefully and always have in mind its limitations. Humans are more than electrons and protons and this calls for the simultaneous use of other ways of thinking - not only observation and induction. There are other ways that also help in searching for the truth. Believing in one thing only can be really dangerous. Believing that only the 'scientific' way of thinking exists can eliminate humanism, morality, and altruism from the world. Many human values are not understandable by “exact science”. More philosophy and less 'scientism' in today’s society of gadgets will prove more than useful in the imminent future.

6. References

  1. Hal Hellman, Great Feuds in Mathematics: Ten of the Liveliest Disputes Ever, Alexandreia publications, Athens, 2010.
  2. Dialethism, Wikipedia article, retrieved from http://en.wikipedia.org/wiki/Dialetheism on 01/08/2011.
  3. Gödel metric: Closed timelike curves, Wikipedia article, retrieved from http://en.wikipedia.org/wiki/G%C3%B6 del_metric# Closed_timelike_curves on 01/08/2011.
  4. Closed timelike curve, Wikipedia article, retrieved from http://en.wikipedia.org/wiki/Closed_timelike_curve on 01/08/2011.
  5. Eternalism (philosophy of time), Wikipedia article, retrieved from http://en.wikipedia.org/wiki/Eternalism_%28philoso phy_of_time%29 on 01/08/2001.
  6. Tim Folger, Time May Not Exist, Discover, June 2007, retrieved from http://discovermagazine.com/2007/jun/in-no-time on 01/08/2011.
  7. Newton's law of universal gravitation, Wikipedia article, retrieved from https://en.wikipedia.org/wiki/Newton's_law_ of_universal_gravitation on 01/08/2011.
  8. Steve Fuller, Dissent Over Descent, 2008.
  9. Dagobert D. Runes, Dictionary of Philosophy, 1942, New York.
  10. Eliminative Materialism, Stanford Encyclopedia of Philosophy, retrieved from https://plato.stanford.edu/entries/materialism-eliminative/ on 01/08/2011.
  11. Physicalism, Stanford Encyclopedia of Philosophy, retrieved from http://plato.stanford.edu/entries/physicalism/ on 01/08/2011.
  12. Bruce Chapman, Materialism's Slipping Hold on Science and Culture, Discovery Institute, Seattle P-I, 1997, retrieved from http://www.discovery.org/a/317 on 01/08/2011.
  13. Roger Trigg, Does Science Need Religion?, Warwick University, retrieved from http://www.st-edmunds.cam.ac.uk/faraday/resources/Faraday%20Papers/Faraday%20Paper%202%20Trigg_EN.pdf on 01/08/2011.
  14. Mediocrity Principle, Wikipedia article, retrieved from https://en.wikipedia.org/wiki/Mediocrity_principle on 01/08/2011.
  15. Copernican Principle, Wikipedia article, retrieved from https://en.wikipedia.org/wiki/Copernican_principle on 01/08/2011.
  16. Hubble, The Observational Approach to Cosmology, Carnegie Institution of Washington. Oxford. At the Clarendon Press. 193
  17. Quotes of dogma from "The Observational Approach to Cosmology", ανακτήθηκαν από: (1) http://nedwww.ipac. caltech.edu/level5/Sept04/Hubble/Hubble3_2.html, (2) http://nedwww.ipac.caltech.edu/level5/Sept04/Hubble/ Hubble3_4.html, (3) http://nedwww.ipac.caltech.edu/level5/ Sept04/Hubble/Hubble3_6.html, The Observational Approach to Cosmology, Edwin Hubble, 1937, p. 50, 51 & 58.
  18. John Polkinghorne, Science or God?, Editions Travlos, Athens, 1996, p. 138.
  19. Paul K. Feyerabend, Farewell to Reason, Editions Ekkremes, Athens, 2002, p. 261.
  20. James DeMeo, Ph.D., Dayton Miller's Ether-Drift Experiments: A Fresh Look, ανακτήθηκε από http://www.orgonelab.org/miller.htm στις 01/09/2011.
  21. Albert Einstein, letter to Besso, cited by Carl Seelig, Zurich, 1954, p. 195.
  22. Popper K., Conjectures and Refutations, p. 142.
  23. Spyridon Kakos, Consciousness and the End of Materialism: Seeking identity and harmony in a dark era, IFIASA, 2018.
  24. List of axioms, Wikipedia article, retrieved from http://en.wikipedia.org/wiki/List_of_axioms on 09/03/2018.

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