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Mathematics is generally & popularly judged a science in the basic duality: science - humanities. As enemies and collaborationists. The border heavily & fiercely policed.

However, it seems to me that Poppers theory, which entitles science-hood by falsification doesn't apply to mathematics at all.

What can it mean that the Number Theory is falsifiable? Certainly a tightly-focused question will either be true or false. More general conjectures & ideas will be true when enunciated as the mathematical landscape is seen and a new shape formed. For example the Langlands programme (higher dimensional representation theory). Significance through aesthetics & ethics seem the key theme. The serious intent (ethic) towards the good & beautiful (aesthetics) towards the reverance & delight of contemplation. Platonism in essence.

Badiou characterises knowledge as four domains (conditions) - love, science, art & politics.

Is it then love - Number Theory being the material incarnation of a mathematicians embrace and adoration of Number?

If not then is it art - Number Theory being the glorification of Number through steady & inspired craft. As a cathedral to God, so Number Theory to the One?

If neither then could it be politics - creating harmony amongst bickering wilful abstract entities intent on having it their way? Number theory being a nation of number systems.

If none of these, then does it lie with Philosophy, the place from which these four conditions converge (in Badious system)? One is reminded that Badiou states that mathematics is the very ontology of philosophy. He may give it space there, but does anyone else?

Geoffrey Thomas
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Mozibur Ullah
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It is the other way around. Falsification is just a statistical form of proof by contradiction. In proof by contradiction, you start off by assuming that a premise p is true. Then you show that such an assumption leads to a conclusion q, which has already been shown to be false. So long as we assume that our axiomatic system is logically consistent, this result implies that p is false: p -> q and ~q together imply ~p.

In falsification/hypothesis testing, we start with a theory T, and then we make an observation O. From T we can determine how likely O is. If the probability of O given T is low, then we can say that the probability of T given O is low. Of course, if the probability of O given T is 0, then we end up with an absolute falsification which would be exactly the same as proof by contradiction.

Of course, in falsification, we rarely have absolutes, and even worse, we are not really testing a single theory but we are using a whole body of theory to come to our conclusion. This result makes since a bit more messy, and I discuss this issue in more detail here.

Daniel Goldman
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After the reconstruction of foundations of mathematics in 20 century, i.e. getting rid of the famous paradoxes (like Russell's paradox), the modern mathematics is based on the belief that no other paradoxes will appear again. Despite numerous efforts, logicians did not manage to prove that the systems of axioms of modern set theories are consistent (and at the same time they did not manage to find new paradoxes). This means that theoretically it is possible that in future somebody will find a new paradox, and this will have the corollary that some mathematical results (possibly most part of them) turn out to be false. This can be considered as the evidence of the falsifiability of mathematics.

Sergei Akbarov
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  • +1:point of information though - Gentzen did actually manage to prove the consistency of arithmetic by assuming stronger axioms. But this doesn't obviate your main claim, I think. – Mozibur Ullah Dec 24 '13 at 20:01
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    No, that cannot be "considered as the evidence of the falsifiability of mathematics". Why do you think that the inconsistency of mathematics is related to falsifiability? – miracle173 Sep 11 '16 at 22:20
  • @miracle173, I explained this. – Sergei Akbarov Sep 12 '16 at 06:51
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    @MoziburUllah, Gentzen's result does not prove the consistency of mathematics. – Sergei Akbarov Sep 25 '16 at 10:13
  • @miracle173 If the theory includes the claim that said mathematics is consistent, and said mathematics turn out to be inconsistent, I think it's reasonable to say that is falsification. – Cort Ammon Aug 21 '18 at 20:14
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    @CortAmmon I am not a native speaker but "I think it's reasonable to say" sounds like that you want to tell me what you associate with the word "falsification" and not how Popper has defined it. Is this right? – miracle173 Aug 21 '18 at 21:14
  • @miracle173 I am using it to point out that what I claim is not a rigorous proof, and appeal to your reason to decide whether to accept or reject it. I am wary of claiming a rigorous proof because mathematics and logic is used to define the language Popper used regarding falsification, and such proofs can run into funny circular arguments if one is not careful. – Cort Ammon Aug 21 '18 at 22:16
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Of course it doesn't apply. It only applies to natural science. And even there, it doesn't apply everywhere. In a strict sense it applies only where you can conduct experiments in a controlled environment, i.e., test hypotheses.

This, IMHO, excludes, for example, the theory of evolution. While this theory does a great job of explaining the development of life, it can not make any predictions. Hence, there is nothing to test and so it doesn't belong to Popper's domain.

Frank Hubeny
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Ingo
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    Some essential parts of the Theory of Evolution can in fact make predictions; for example the famous Mendel Laws of genetics. There are also numerous related controlled experiments on the species that reproduce fast enough to observe numerous generations. Finally, there were numerous predictions of what kind of fossils would be discovered, and many of those predictions were confirmed by later digs. – Michael Dec 26 '13 at 02:57
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    @Michael IMHO, the Mendel Laws could also hold in a Creationist Setting. Is it not rather so that evolution works by violating Mendels Laws (through making DNA replication errors) from time to time. Also, in a strong sense, predictions of fossils, while being great, are not quite on par with controlled experiments. One could ask: How many predictions were not fulfilled? Can I not make 1000 predictions, but then, when 1 comes through, I claim scientific victory? – Ingo Dec 26 '13 at 13:15
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    I do not agree with you; see from [Imre Lakatos](http://en.wikipedia.org/wiki/Imre_Lakatos) (he studied with Popper) : *Proofs and Refutations* (1976) : "was a Hungarian philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its 'methodology of proofs and refutations' in its pre-axiomatic stages of development, and also for introducing the concept of the 'research programme' in his methodology of scientific research programmes." – Mauro ALLEGRANZA Apr 02 '14 at 10:35
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    "Fallibility of mathematics"? Then I know I do not need to read this author, as he has no understanding, obviously. – Ingo Apr 02 '14 at 14:21
  • @Ingo: Mathematicians do mathematics using physical objects, e.g. - paper, pen, the mathematician's brain. So their knowledge of mathematics is entirely dependent on their knowledge of the physical behaviour of those objects, and is just as fallible as their knowledge of those objects. See "The Fabric of Reality" by David Deutsch Chapter 10 for more details. – alanf Jan 02 '16 at 11:44
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    @alanf I consider this a very lame argumentation. First, let us forget paper and pencil, they're not really needed. The brain certainly is. But it is simply not the case that ones mathematical knowledge depends on ones understanding of the function of the brain. To the contrary, one can make most important mathematical discoveries(!) without even a basic understanding of the brain. So, refuted ... – Ingo Jan 02 '16 at 15:27
  • @Ingo You presume that the brain is not falliable, or that one's mathematical knowledge is stored elsewhere besides the brain. That's actually quite a substantial assumption to start from. It can also be used to defend many other things besides mathematics, such as the religion of your choosing. – Cort Ammon Jan 02 '16 at 19:33
  • @CortAmmon I presume nothing of that sort. Nor do I deny that mathematicians can commit errors, err in proofs, etc. But to put some flesh on the bone, maybe you Popperians tell us how to falsify the truth that there are infinitely many prime numbers. Just to start with some simple theorem that is (together with its proof) well known since thousands of years. I bet you can't even devise a strategy how to falsify it. The reason is that it is simply not falsifyable, because pure Math is not in Poppers domain. – Ingo Jan 02 '16 at 22:06
  • @Ingo You're right, I cant falsify the proof that there are infinitely many prime numbers, using the axioms that you are presuming we all use. I would probably start from the finitist set of assumptions. You do know that not everybody agrees on which axioms should be at the bottom of mathematics, right? We might even need to develop tests to see which axioms make the most sense for the world we live in. – Cort Ammon Jan 02 '16 at 23:10
  • Besides, mathematics fails people all the time. They certainly failed Bertrand Russel with his Principia Mathematica. He was so very confident that he'd found the mathematics he wanted, only to find that the mathematics in his head was not proven out on paper. – Cort Ammon Jan 02 '16 at 23:12
  • Another potential place where mathematics fails. The human imagination has no problem working with a set that contains itself as element. Its only when you apply modern set theory that it starts to fall apart, so much so that many choose to add self-referential sets to set theory and try to create theories that can align with both approaches. – Cort Ammon Jan 02 '16 at 23:14
  • @Ingo The idea that your thoughts can and do obey the laws of logic or arithmetic or whatever when you're thinking about maths requires making substantive assumptions about how thoughts are instantiated, i.e. - assumptions about the brain. Also, Popper's epistemology is not just about science and falsification. More generally, all knowledge is created by proposing solutions to problems, and criticising the solutions until only one is left and it has no known problems. – alanf Jan 02 '16 at 23:52
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    @CortAmmon I am quite sure you confuse Russell and Frege. But again, nobody says that mathematicians cannot err. The so-called finitists are widely regarded as cranks, trying to prove all their life that "Cantor erred". – Ingo Jan 03 '16 at 09:21
  • @alanf I would put it differently: from the fact that we do make mathematical discoveries, we can certainly conclude something about the brain. This latter knowledge is of course in Poppers realm. Your latter sentence of course does apply to math, yet isn't it kind of trivial? The OP asked specifically about "Falsification", and here we have basic epistemologic problems when applying it to math, because many math theorems claim the existence of something, and it is impossible to empirically refute such claims. – Ingo Jan 03 '16 at 09:30
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    Creationist alert. The theory of evolution makes loads of predictions. – Joe Lee-Doktor Aug 21 '18 at 16:54
  • @Ingo. Mathematics often fails to represent a relevant reality. For instance in Proofs and Refutations, which traces the history of Euler's theorem on polygons, there is an ongoing issue of whether a 'star' (as in the inside of a pentagram) is a polygon. Whether Euler's theorem is proven by various proofs, including his own, depends on that consideration. If the math does not match our natural assumptions, I would say it fails, because it is supposed to formalize just those assumptions. We don't pick axioms at random, we pick them to capture some real idea, and they can therefore be wrong. –  Aug 22 '18 at 16:31
  • That does not imply that mathematics has relevance primarily to the degree it risks having its axioms be wrong -- that is just the most direct way of failing to capture relevant thinking. But every elaboration is more valuable if it risks being pointless, than if it safely reiterates known patterns that could pretty much just be worked out by a machine, and will be worked out by technicians in other fields as needed. You don't need mathematicians to do those things... –  Aug 22 '18 at 17:02
  • @JoeLee-Doktor I'm *not* a creationist at all (actually, I am atheist). That being said, I don't doubt there are predictions. But can you point out (or point to) some readily testable hypotheses? – Ingo Aug 24 '18 at 22:24
  • @jobermark the history of science, especially physics, is full of seemingly "irrelevant" mathematical results and techniques that suddenly became most important. Mark this: just because **now** nobody in the world understands what this crazy mathematician is talking about doesn't mean this will be so a couple of decades later. – Ingo Aug 24 '18 at 22:30
  • @Ingo I think that totally misses the point of what I said. Popper's criterion is about taking the *risk* of falsifiability, not about actual falsification. –  Aug 24 '18 at 22:59
  • Ok, @jobermark and I defend the point that math takes no such risk because an actual falsification is impossible. The point is that when you find an error in a proof you haven't falsified math, but only the (implicit) claim that this proof is proper math. Or if you show that some system of axioms is inconsistent, you haven't falsified math, but only the idea that any theorems derived from such axioms were mathematical ones. If it's demonstrably false, then it isn't math. Simple as that. – Ingo Aug 24 '18 at 23:16
  • @Ingo And not everyone agrees, including Lakatos, whom you refuse to read, so I am giving you part of his argument. Still, responses should be responses to what is actually said, not outright non-sequiturs. You don't have to agree with me. But when you say stuff that obviously misses my point entirely, we are not in the same conversation and you don't care to listen. I will go away now and avoid responding to someone who does not care what I have to say... –  Aug 24 '18 at 23:31
  • @jobermark one important point of mathematics is to avoid stupid searches for the "right" number by proving that it cannot possibly exist. OTOH, there are theorems that prove the existence of something. For example, if p is prime, then there exists a number q that is prime and greater than p. Find a "counterexample" for that, it doesn't even remotely make sense. You cannot empirically falsify an existence theorem. – Ingo Aug 25 '18 at 00:25
  • @Ingo And I have not said one can. So why are you addressing this contradiction of something I never said to me? If you don't even really read what I wrote, just don't respond. You are being a hypocrite who thinks we need to listen to you, while you fail to actually listen to anyone else. –  Aug 25 '18 at 00:27
  • @jobermark There is a comment missing from me, where I cited Lakatos who held the view "... that we should not think that a theorem is ultimately true, only that no counterexample has yet been found." (according to Wikipedia). And you suggested Lakatos as authority on fallability of math. So I came up with 2 scenarios where "counterexamples" are silly or entirely meaningless. – Ingo Aug 25 '18 at 00:43
  • @Ingo Yes, abusive comments are deleted. But I am not making the point, Lakatos didn't either. Wikipedia is just unreliable. You have dismissed what I said and are now objecting to things I did not say. And you are being obnoxious about it. You have some theory of what math really is, no doubt something pseudoreligious like Platonism. And you reject the notion that math is psychological and discovered by exposing ideas to the mind by trial and error. You reject my position. Fine. Now go away. –  Aug 25 '18 at 02:19
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    @Ingo The irony here is that Darwin formulated concept of falsifiability long before Popper: "If it could be demonstrated that any complex organ existed which could not possibly have been formed by numerous, successive, slight modifications, my theory would absolutely break down" – IMil Nov 13 '19 at 00:23
  • @IMil This is interesting, however, it is unclear how such a demonstration could possibly (sic!) work. Anyway, I may be mistaken about the epistemological status of evolution theory, which I regard highly, despite I still think that it's different than other natural sciences. – Ingo Nov 13 '19 at 19:58
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    @Ingo oh, there are plenty of possibilities. In our times, with genetic engineering, we routinely create living creatures that could not have evolved naturally. Well, some rudimentary horizontal gene transfer occurs naturally, but if a sizeable portion of DNA from an antarctic fish just happens to be present in a totally unrelated Canadian cucumber variety, this strongly suggests that there has been some artificial process involved. – IMil Nov 13 '19 at 23:54
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    @Ingo but even in regard to Darwin's time, you seem to commit a slight fallacy. You seem to think along the lines "well, if the biologists have a good imagination, they could always come up with an evolutionary explanation for any organ". But in fact, it's the other way round. Biologists, since Darwin, were always able to eventually come up with an evolutionary explanation for any organ or species precisely because this theory is true. In an intelligently-designed world, there would be infinite possibilities for organs or species with no evolutionary explanation. – IMil Nov 13 '19 at 23:59
  • @IMil It's too bad we have this discussion in constrained space and a tiny font I can barely read anymore. However: I do not dispute that Darwin's theory is true. To the contrary, I strongly think it is true. Nevertheless, falsifiable predictions and controlled experiments simply don't happen in this science. (I think, it's different in genetics, though. But that genetics works is no proof for Darwin, it just makes it even more plausible by pointing out the possible mechanisms at work.) – Ingo Nov 27 '19 at 17:39
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Absolutely the concept of falsifiability applies to mathematics (at least in a roughly metaphorical way, and maybe much more). I am not a big expert on Popper but ever since hearing it, I think it is possibly more relevant/applicable to math than almost any other science/field in the following sense.

Math works through the advance of hypothesize, test, and verify. In this way it is no different than any other scientific field, and in many ways embodies all the others in a more pure form. The hypothesis is typically called a "conjecture". There are many open conjectures, some quite famous, and very old. (e.g., there are Greek number theory problems over 2 millennia old), but a single counterexample is a falsification of a conjecture. Notice that one does not need to do complex experiments to refute or replicate a falsifiable claim. It's all abstract, cerebral/mental.

However, there is also nowadays even an experimental/empirical falsifiability going on in math, in increasingly widespread/significant use: computers are used to explore conjectures and find "not too big" counterexamples.

Note, however, there are some senses where falsifiability fails in math. undecidable problems do not have algorithmic answers (e.g., Hilbert's 10th problem, undecidability of finding solutions to diophantine polynomials). (In fact it's striking how many parallels there are in the theory of falsifiability to undecidability.)

Some theorems are unprovable. Some problems are extremely difficult and may never be resolved, (e.g., the Riemann conjecture open 1.5 centuries even after intense focus), etc. Also, the issue of replicability of falsifiability rears its head with computer assisted proofs, e.g., the famous 4 color theorem. Can they be trusted?

It is tempting to propose that all well-formed math conjectures are falsifiable and Hilbert had such a belief early on and wrote about it and the idea motivated some of his own conjectures, but of course Godel strikingly proved otherwise.

Another issue that arises is "proofs that were later found to be incorrect" (Mathoverflow). So human fallibility and falsifiability are interrelated. Anyway one might say that mathematicians are more rigorous about and dedicated to falsifiability than any other "scientists".

Frank Hubeny
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vzn
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  • somewhat related [is mathematics always correct?](http://philosophy.stackexchange.com/questions/3716/is-mathematics-always-correct) – vzn Mar 12 '14 at 04:19
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    vzn, Popperian falsifiability is **not** the same concept as "proof by contradiction" in logic and mathematics. – robert bristow-johnson Jan 02 '16 at 19:33
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    -1, "I am not a big expert on Popper": It is not a good idea to answer the question without knowing the meaning of the terminology and the basics of Popper's thinking. robert bristow-johnson already wrote in a comment that your interpretation of falsifiability is wrong. – miracle173 Sep 24 '16 at 10:49
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although often called "The Language of Science", Mathematics is not the same as Science (or, more specifically, Natural Science).

Mathematics is a specific application of philosophy and logic to the concept of quantity. whereas in Science that quantity of "stuff", whatever that stuff is, is salient, the techniques of mathematics is necessary to understand more fully what the quantities are, how they may be related, and to speculate in how some quantities may be related to others. in that, there are predictions of how specific physical or natural quantities will behave, experiments devised, observations made, and these predictions are supported or falsified. scientific theories can be developed and supported from that.

but there is no such counterpart in pure mathematics. sometimes conjectures are made (such as Bertrand's postulate or Fermat's Last Theorem), but they are just conjectures until they are proved by derivation, not by observation as is done in science.

now these two conjectures, now proven mathematical facts will never be disproved in the future. that is also different than falsifiability in science. General Relativity at this time seems like the unrefuted truth, but in another time, so was Newtonian mechanics and gravitation, but now we understand that to be a very close approximation to the truth for slow relative speeds and not in extremely intense gravitational fields. so nothing in Science has the status of never, ever, being falsified.

but in mathematics, a properly proven theorem will never be disproven given the same axioms to the theorem.

it's different. mathematics and natural science are not the same things. mathematics, although a salient tool in science, is not science. Popperian falsifiability applies to science. it is a demarcation of ultimately what is science and what is not. not all scientists agree with that demarcation, but i do.

robert bristow-johnson
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    +1 I think simply said mathematics generates tautologies and tautologies can't be falsified. – miracle173 Sep 11 '16 at 23:00
  • @miracle173, you should prove the consistency of mathematics for being convincing in this point: https://en.wikipedia.org/wiki/Consistency – Sergei Akbarov Sep 12 '16 at 16:28
  • @robert bristow-johnson You claim that "Although often called "The Language of Science", Mathematics is not the same as Science (or, more specifically, Natural Science)." Do you have any specific reason to suggest that science is a synonym of natural science? As far as I'm concerned, both natural and social sciences are legit parts of science. Try to suggest in an academic article that linguistics, psychology or economics are completely non-scientific like mathemathics. You will not succeed. – Ana López Nov 29 '17 at 12:42
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    i don't think that i suggested that linguistics, psychology, or economics are not science. *"science"* means, fundamentally, *"knowledge"*. and while scientific method can *often* (not always) be applied to those fields, there is something that these *soft sciences* have that elude consistent results in their experimentation. it's different than natural sciences that do not depend of the capricious whims of the human mind, although often we can get consistent results in the sense of an ensemble or statistical average. – robert bristow-johnson Dec 01 '17 at 00:48
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Mathematics is a branch of metaphysics according to Popper, see "Realism and the Aim of Science" Part I Chapter III. Popper does not judge metaphysics adversely compared to science. Some people want to draw that implication because they uncritically admire everything labelled science as a substitute for making their own judgements.

For a couple of other perspectives on maths that seem broadly consistent with Popper's epistemology, see "The Fabric of Reality" by David Deutsch, Chapter 10 and "Proofs and Refutations" by Lakatos. As a side note, I do not recommend any other material written by Lakatos, who wrote a lot of material attributing positions to Popper that he never advocated.

alanf
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As a corollary to some of the other answers here, it is worth noting that whether you consider mathematics to be falsifiable is likely to depend on your account of what mathematics is.

On a formalist account, mathematicians are just manipulating strings of symbols according to rules. As such, the mathematics itself cannot be falsified, although there can always be mistakes that need to be rectified. On this view, it is only when the strings are given an interpretation that a falsifiable proposition emerges.

On an empiricist account, the question of whether mathematics is falsifiable goes hand in glove with the question of whether logic is empirical. This is a question on which a lot has been written. Famously, Quine held that all of our knowledge, including logic, is holistic in nature and falsifiable only as a whole, not on a sentence by sentence basis. Consequently, even logical or mathematical statements are revisable in the light of future knowledge - they have no special a priori status - they are just relatively well protected from being revised because they serve an important and central role in organising our web of belief. Also, Putnam wrote an influential paper called "Is Logic Empirical?" in which he proposed that we might want to revise our logic in the light of quantum mechanics.

On a logicist view, mathematics is reducible to logic, so a mathematical statement is falsifiable only in the sense that it might be possible to show that it entails a logical inconsistency.

To an intuitionist, a mathematical proposition is something constructed from the mathematician's intuition. One might therefore reasonably suppose that mathematics is falsifable to just the same degree that a mathematician's intuition is falsifiable.

As to mathematical platonism: I'm not sure that I really understand it, but I suppose that platonists would regard mathematics as falsifiable, just not empirically.

Bumble
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  • AFAIK, mathematical platonism holds that there is a "third realm" (besides the realms of matter and ideas) which is inhabitated by mathematical laws and object. Or, to put it differently, that mathematical objects and laws exist independently of human consciousness. The platonist mathematician merely "discovers" this realm. Hence no falsification possible. – Ingo Aug 24 '18 at 22:14
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I will continue to argue my tiny minority position that mathematics is a branch of psychology.

Mathematics is still science. It is the science that tries to identify the thought processes and forms of idealization that humans share, that improve our leverage upon thought and science.

Mathematical disciplines do not suffer from conflict with the facts, but they clearly descend into irrelevance when the facts they apply to do not matter. What is false, when math is 'falsified' and becomes irrelevant, is the idea that a given question is interesting to the world at large, or has use because it captures an important insight.

  • It is interesting that (one of) the father(s) of modern mathematics, Gottlob Frege, destroyed the "mathematics is a branch of psychology" idea in the 19th century. And rightly so, since psychology is not even natural science at all. – Ingo Aug 24 '18 at 23:24
  • @Ingo Your bigotry is not welcome here. Dismissing psychology as a science is not logic, it is pointless offensiveness. And I can dismiss Frege with actual evidence. He was brilliant and very close to the truth, but ultimately just wrong, and his project needed to be started over from the ground up. –  Aug 24 '18 at 23:36
  • Interesting that this reported destruction comes with patent smugness and no evidence. Having been wrong about his life's work, he could be wrong about other things as well. –  Aug 24 '18 at 23:57
  • The epistemologic problem with psychology is its subject matter. Humans can learn (this you cannot deny, obviously) hence any experiments you make can have a different outcome next time. Or, to put it differently, because humans can and do learn, the usual "under same circumstances" assumption never holds. So it is questionable if psychological knowledge is indeed scientific knowledge proper. – Ingo Aug 25 '18 at 00:09
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    @Ingo Dogs can also learn, and yet zoology is a science, and animal behavior is properly included therein. This is not a deduction, it is shoring up a bias. And learning is just the way minds change. The subjects of most other sciences also change in irreversible ways. You might as well say that meteorology is not a science because the world is never again in the same state it was yesterday. Or that evolution is unscientific because we cannot reverse time and re run the generation of a species again and again. –  Aug 25 '18 at 00:14
  • The difference is, objects of other sciences cannot learn how to spoil the outcomes of the experiments they undergo. Take this famous experiment where the objects were encouraged to torture a human with electric shocks, and they did. It is entirely possible that humankind has learned enough so that this experiment can never be repeated. (I for my part would certainly **not** press the button.) – Ingo Aug 25 '18 at 00:35
  • Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/82211/discussion-between-jobermark-and-ingo). –  Aug 25 '18 at 02:23
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Since Theorems in Mathematics are those formulas that are proved and every proved formula is true (Completeness of 1st order logic), the falsifiability can not be applied to Theorems in Mathematics.

Esfandiar Eslami
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