14 ms·
Has philosophy ever clarified mathematics?
- zdean 10y agoWouldn't mathematics simply be considered an epistemological branch of philosophy?
- ebola1717 10y agoNo, not at all. One could argue that analytic philosophy and mathematical logic have some overlap and inform each other, but even then, there are deep differences in their goals and methods.
- thwd 10y agoThere's a joke: Mathematics is just applied Philosophy. It bears some truth :)
- dnautics 10y agoI've thought of that more of a serious thought than a joke. At my college we had a joke that physics majors washed out and became math majors (because the faculty were assholes); math majors washed out and became philosophy majors (because they couldn't handle rigor); and philosophy majors washed out and became anthropology majors (because they couldn't handle homework)
- mannykannot 10y agoTalk of math, philosophy and jokes reminded me of this post: https://news.ycombinator.com/item?id=13610134 https://news.ycombinator.com/item?id=13610134
- baq 10y agothe top answer really doesn't leave much to argue about.
- bykovich 10y agoThe top answer is extremely wrong.
- grandalf 10y agoIsn't the crux of this back and forth proofs that rely upon the axiom of choice vs those that do not? It is my understanding that philosophers have added a lot to our understanding of how the axiom of choice impacts logical reasoning about things which matter to humanity in concrete ways.
- gpawl 10y agoWhat proofs? Axiom of Choice cannot matter to humanity in concrete ways (except as an artistic activity), because the Axiom of Choice only applies to uncountably-large sets, which have no realizable physical meaning in the Universe.
- raverbashing 10y agoIt's an interesting question However while there are no physical realizations of infinite sets, they're used all the time in math So I guess mathematical entities do not need to be connected to physical elements
- evincarofautumn 10y agoOf course. It’s not clear that the “real” or even “natural” numbers meaningfully correspond to any physical phenomenon. They’re just abstractions and tools for making predictions and reasoning about our experiences as humans, which don’t necessarily reflect how the world actually is.
- grandalf 10y agohttps://arxiv.org/pdf/1401.6574.pdf https://arxiv.org/pdf/1401.6574.pdf
- colorint 10y agoThe universe is uncountably infinite; that people routinely stumble over this is why Zeno posed all those paradoxes about motion. In a sense, if you don't believe that reality is uncountably infinite, then you believe it IS possible to square the circle. This actually matters a great deal, since it impacts on things like, what are digital computers (which is to say, set theoretic theories of math) capable of?
- platz 10y agoCarlo Rovelli: "Why Physics needs Philosophy" https://www.youtube.com/watch?v=IJ0uPkG-pr4 https://www.youtube.com/watch?v=IJ0uPkG-pr4 A few notes from the above: * the beginning of astronomy == plato's school * the scientific method as falsification == Popper * quantum theory / relativity / Heisenberg == positivism (If I don't see it (e.g. electron orbitals) it doesn't exist) (* e.g. complementarity) * Einstein claimed that his reading of Schopenhauer was crucial to thinking about time, space, etc... in essence, you are doing philosophy when you're re-evaluating your methodology and using a evolving reflective feedback loops to change your thinking.
- euyyn 10y ago> quantum theory / relativity / Heisenberg == positivism (If I don't see it (e.g. electron orbitals) it doesn't exist) Wow, that is such a non-sequitur that it casts doubt on the whole thing.
- platz 10y agoyes, it's an oversimplification, and probably a slight misread of the point the guy was making.. i don't have time to copy the whole talk into a single sentance. a more charitable way to think about it is to consider the effects positivism had on scientists at that time.
- euyyn 10y agoAh, that makes sense.
- mannykannot 10y agoFurthermore, I believe there was quite a lot of physics done before Popper explained how to do it. He said many valid and insightful things, but it was not necessary for physics for him to say them.
- platz 10y agoPopper didn't explain "how to do physics", nor did he invalidate previous physics, nor did he "invent the scientific method". But would you not agree that he influenced how scientists interpreted the scientific method? This is the point I'm making.. please don't reduce it into a blanket idea such as "phsyics could not have been done before Popper" which isn't anywhere close to what I said.
- bykovich 10y ago> Secondly, remember that broadly the point of philosophy is to make things not philosophy. In extremely simplistic historical terms, once natural philosophy becomes rigorous it becomes science, once philosophy of language became rigorous it became linguistics, and today we're seeing philosophy of mind turn to neuroscience. This is absolutely not correct, and elucidates little but the prejudices of the answerer. Philosophy of language has /not/ become linguistics, philosophy of mind has /not/ become neuroscience, and only a subset of natural philosophy has become natural science. The philosophical questions discussed by Socrates have elided the grasp of both dogmatic rigor and empiricism for twenty-four hundred years, and there seems to be absolutely no reason to expect this to change. The answerer has either no actual grasp of the history or content of philosophy, or has simply decided, apparently by fiat, to discard all but the most narrow positivism-flavored slice as nonsense.
- darklajid 10y agoYou seem to have a certain grasp on that subject or at least feel very strong about it. Unfortunately your post didn't correct any (perceived) mistake, it was mostly a longer way to write "That's not true". Would you be so kind and expand that reply a bit, so that I might find pointers/key words to look up myself?
- dcre 10y agoAll that's required to show that philosophy's aim is not to eliminate itself are a few examples of philosophical questions that are not reducible to scientific ones. Naturally the author of the quoted post could still argue that those examples are in fact reducible. They'd be wrong, but I'm sure they could attempt an argument. That's philosophy. Here are some articles from the Stanford Encyclopedia of Philosophy on the mentioned topics that should point you in the right direction to start thinking for yourself about whether these questions are reducible to the empirical sciences: Natural philosophy Aristotle's natural philosophy: https://plato.stanford.edu/entries/aristotle-natphil/ https://plato.stanford.edu/entries/aristotle-natphil/ Hume (look at the section on causation): https://plato.stanford.edu/entries/hume/ https://plato.stanford.edu/entries/hume/ Philosophy of language Wittgenstein: https://plato.stanford.edu/entries/wittgenstein/ https://plato.stanford.edu/entries/wittgenstein/ Names: https://plato.stanford.edu/entries/names/ https://plato.stanford.edu/entries/names/ Philosophy of mind Aristotle's psychology: https://plato.stanford.edu/entries/aristotle-psychology/ https://plato.stanford.edu/entries/aristotle-psychology/ Kant's account of reason: https://plato.stanford.edu/entries/kant-reason/ https://plato.stanford.edu/entries/kant-reason/
- CalChris 10y agoBishop Berkeley's response to Newton might qualify. Newton had something useful but Berkeley showed he hadn't proved anything. It wasn't until Cauchy (?) proved things rigorously that calculus was on a solid footing.
- platz 10y agoAlso, philosophy has been dogged since Descartes with the self-imposed goal of providing a "foundation" for all knowledge enterprises (e.g. science, math) — by attempting to, put crudely, solve the mind-body problem and legitimatize that our thoughts and connection to the world are valid.
- woodrowbarlow 10y agoOn the flip side, here's an example of mathematicians clarifying philosophy. https://en.m.wikipedia.org/wiki/Mike_Alder#Newton.27s_flaming_laser_sword https://en.m.wikipedia.org/wiki/Mike_Alder#Newton.27s_flamin...
- scythe 10y agoIn some sense Grothendieck's investigations could be considered "philosophical"; in the early 20th century algebraic geometers studied objects called "varieties" and Grothendieck's coup resulted from asking the question "what is the general class of object with which we can do algebraic geometry?". Today scheme theory is entirely mathematical, but a scheme had to be conceived as a philosophical concept first. The link also mentions Turing's elucidation of the Turing machine as a similar process. There is also the case of Frank Ramsey and Piero Sraffa, who were the only close friends of Ludwig Wittgenstein, and who went on to make major epistemological contributions to economics (and Ramsey was a philosopher in his own right): Ramsey was the first person to really clarify the concept of a subjective probability, and Sraffa was central in the capital aggregation controversy.
- pron 10y agoI am not familiar with Grothendieck's work and with algebraic geometry in general, but Turing's work was rather unique, and very much philosophical in nature[0]. I am basing the following on Turing's 1936 paper, On Computable Numbers[1], on Juliet Floyd's chapter on Turing's mathematical philosophy[2], and on Robin Gandy's 1988 paper, The Confluence of Ideas in 1936[3]. The generalization of mathematical concepts has always been an important part of mathematics. In fact, Gandy writes that this was precisely the concern with Church's claim that the lambda calculus can express all algorithms ("a vague and intuitive notion", in Church's words). It was thought that perhaps a genius could some day generalize the notion further to include lambda-definability as well as things that fall outside it. But then Turing appeared with arguments of a very different kind. For one, his construction of an automatic machine, while precise, was completely arbitrary. It was clear to everyone from his treatment that while his math concerns a specific formalism (what's known as the Turing machine ever since Church named it so), it really covers all formalisms. For another, his proof of undecidability is very different from the one normally presented as the proof for the undecidability of the halting problem (which is really due to Martin Davis). Turing's proof, Floyd notes, makes no use at all of any logical construction, and does not rely even on logical contradiction or the law of excluded middle (which are expected to hold in many "reasonable" formalisms) but on more basic philosophical principles that must hold in any and all formalisms. Gödel called Turing's achievement a "miracle". In 1946 he said (quoted in Gandy's paper): ...with this concept one has for the first time succeeded in giving an absolute definition of an interesting epistemological notion, i.e., one not depending on the formalism chosen. In all other cases treated previously... one has been able to define them only relative to a given language, and for each individual language it is clear that the one thus obtained is not the one looked for. For the concept of computability, however,... the situation is different. By a kind of miracle it is not necessary to distinguish orders, and the diagonal procedure does not lead outside the defined notion... [0]: Turing was a mathematical philosopher among other things, and is known for his discussions with Wittgenstein, published in Wittgenstein's Lectures on the Foundations of Mathematics. [1]: https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf [2]: https://mdetlefsen.nd.edu/assets/201037/jf.turing.pdf https://mdetlefsen.nd.edu/assets/201037/jf.turing.pdf to appear in Philosophical Explorations of the Legacy of Alan Turing [3]: http://dl.acm.org/citation.cfm?id=213993 http://dl.acm.org/citation.cfm?id=213993 published in The Universal Turing Machine A Half Century Survey, 1995
- PravlageTiem 10y agoEpistemology for everyone! Truth is the intersection of belief and truth. The moment you start formalizing a belief, be it linguistically, mathematically, naturally, structurally, etc., you are no longer talking about the belief, but the model of that belief. What we consider axiomatically true in the domain of logic could just be artifacts of a confirmation bias between the structure of our brains and the patterns we have evolved to seek. (Facial expressions, social cues, emotions, etc.) In this sense, logic appears to reveal itself to us, when it reality we are inventing an approximation of it in accordance to the real hard limitations of our neurology. Philosophy, then, is about unwrapping the truth from belief to get down to the core BELIEF, not the core TRUTH. Beetles in an Austrianian desert almost went extinct because their axiomatic truth about brown, shiny, and rigid being ideal for mating was completely hijacked by empty beer bottles.
- mymythisisthis 10y agoMost of the famous mathematicians were driven by philosophy. For example Kepler wanted to keep perfection in the new model universe. He thought he could do it by using Plato's solids for the relative distances between the planets. He was wrong, but the mathematical attempts he tired help to form a more correct answer. The motivation was philosophical.
- DanBC 10y agoIt might be interesting to ask the same question on the partner site http://philosophy.stackexchange.com/ http://philosophy.stackexchange.com/
- elangelcentral 10y agoYep https://en.wikipedia.org/wiki/Russell's_paradox https://en.wikipedia.org/wiki/Russell's_paradox
- philofcompguy 10y agoAs you may know, many mathematicians turned philosophers while trying to do work on the foundations of mathematics. It seems like logic was the gateway discipline. What we now know as the analytic turn in philosophy of early 20th century came from such as lineage before it devolved into philosophy of language. Frege, Russel, Whitehead, and Wittgenstein are well known in this line of thought. However, the history of philosophy and mathematics goes way back to pre-socratic philosophers like Pythagoras and centuries later Aquinas and then Descartes. The question posed is rather strange given that mathematical development has often been formed by philosophical thought. I guess by clarifying the author means providing solutions (since he/she mentions that "mathematical insight" is nowhere to be found in the literature)? Since philosophy is not the same as doing mathematics the only kinds of clarification that philosophy will provide is in terms of distinctions, definitions and criteria: what is a proof, etc. This is because 'philosophy of x' is always meta discipline.
- ClayFerguson 10y agoI think it's quite safe to say philosophy has never contributed anything whatsoever to any kind of math, science, or engineering. Philosophy is all about helping mankind understand how to think about thought itself. Also I wouldn't call psychology a science either. Until mankind has a clue what consciousness actually is, there will be an impenetrable firewall between philosophy and all sciences.
- unit91 10y ago> Question: Has it ever happened that philosophy has elucidated and clarified a mathematical concept, proof, or construction in a way useful to research mathematicians? I would hope so! The short answer is the philosophy of Math will help you determine whether what you're researching is true! Surely it would be very bizarre to research something with complete apathy regarding its truth value. A few examples: The famous Peano axioms [1] are widely used to prove such things as the commutative property of multiplication (ab=ba). But as the name "axiom" suggests, you just have to accept them as true or the whole thing crumbles. So why is it true that "0 is a natural number"? If this is false, much (all?) of math research is in big trouble! Does this suggest a sort of mathematical epistemic foundationalism? If so, what are its limits? When is mathematical research warranted, and when can we simply regard mathematical beliefs as properly basic? Also, consider the realist/anti-realist debate [2, 3] which seeks to answer the question "are numbers, sets, functions, etc. actual features of the real world, or are they all just in our heads?" (or some refined variation thereof). If they are real entities, how is it that these non-causal things (like 5) lie at the very heart of the laws governing the physical, causal universe? But if they aren't real, then what possible explanation can one give for the perfect harmony of the physical world and these functions, that are ultimately all in my head? Moreover, why is belief in these unreal entities so widespread (I know of no "amathists")? [1] https://en.wikipedia.org/wiki/Peano_axioms https://en.wikipedia.org/wiki/Peano_axioms [2] https://plato.stanford.edu/entries/platonism-mathematics/ https://plato.stanford.edu/entries/platonism-mathematics/ [3] https://plato.stanford.edu/entries/scientific-realism/ https://plato.stanford.edu/entries/scientific-realism/
- gizmo686 10y ago>So why is it true that "0 is a natural number"? If this is false, much (all?) of math research is in big trouble! Does this suggest a sort of mathematical epistemic foundationalism? Math major here. To me, this seems like a non-nonsensical question. 0 is defined to be a natural number. You could say (as Platonism does), that there is some "real" natural numbers out there in the universe, and ask if the Peano axioms define a system that is isomorphic to the real "natural numbers". Or you could ask if the Peano axioms acuratly define the artificial system that we informally think of as the natural numbers. You could also ask if any system (real or abstract) can satisfy the Peano axioms. That is to say, are they internally consistent (and is there a constructive proof of such). Assuming this is the case, then (by definition) there will be an object that we call "0" that is a natural number. You can also ask if there is a unique (up to isomorphism) instantiation of the Peano axioms (there is).
- balsam 10y agophilosophy of philosophy, a godelian knot.
- balsam 10y agophilosophy of philosophy, a godelian-gordian knot
- Entangled 10y agoYes, mathematics is just an unidirectional arrow of opinionated integers. And 42.
- decasteve 10y agoBill Lawvere was heavily influenced by philosophy in his [revolutionary] contributions to the development of Category Theory [1][2]. [1] https://ncatlab.org/nlab/show/William%20Lawvere#RelationToPhilosophy https://ncatlab.org/nlab/show/William%20Lawvere#RelationToPh... [2] http://philosophy.stackexchange.com/questions/9768/have-professional-philosophers-contributed-to-other-fields-in-the-last-20-years/9814#9814 http://philosophy.stackexchange.com/questions/9768/have-prof...
- voidhorse 10y agoFrege, Russell, Whitehead, Pierce--of course! It doesn't help that many philosophers of mathematics are, for obvious reasons, either also logicians or mathematicians, so demarcating between advancements in philosophy of mathematics that clarify mathematics and advancements in mathematics that clarify mathematics can be a bit of a fool's errand. Whatever the case, I dislike it when folks from the sciences or mathematics try to discredit or dismiss philosophy--funnier still, and luckily not as bad, is when they question the value of philosophy without realizing that question is in and of itself a highly philosophical question! Philosophy has been around for a long time and isn't going anywhere in the perceivable future (though I suppose it depends on what metaphysics of time you subscribe to :) ).
- Mendenhall 10y agohmmm I would say philosophy "clarifies" where mathematics comes from.
- convolvatron 10y agoif you asked practicing mathematicians whether or not the philosophy of foundational mathematics, the hilbert programme, and wittgensteins consternation on the notion of equality were at all relevant...you would get a pretty dispassionate 'of course'. for whatever reasons most of these topics are taught in the philosophy departments and not the mathematics.
- mikehain 10y agoPhilosophy often revolves around questions of purpose in life, and any non-mathematician might wonder why someone would devote their life to the study of advanced math. The question of purpose becomes even more interesting when a mathematician spends decades solving theoretical math problems that have no immediate practical use in engineering or programming. Any mathematician who's engaged in such problems wouldn't have any trouble understanding his own motives; for him, the philosophy is so obvious that it goes without saying. But an outsider might see his actions as inexplicable - as inexplicable as an ascetic monk spending a week meditating on a frigid mountaintop. Philosophy functions as an aid in that understanding. The mathematician might read it and recognize truth, if it is written well, and a non-mathematician might be able to learn the rationale for devoting one's life to math. It might have a strong enough effect to make someone gravitate towards that type of life. Different works in the philosophy of math could deal with different aspects of the mathematical life, such as the inherent beauty of an elegant proof, and it could also go into the larger, more long-term purposes of math, whose effects might not be recognized until long after the mathematician is dead. I would think that the best philosophy of math would need to be written by a mathematician. An issue with philosophy is that it's quite difficult to write it well. There is a lot of philosophy out there that's poorly-written. It might suffer from a lack of clarity or simply be inaccurate. To have a brilliant philosophy of math, you would need a person who is both a skilled mathematician and a skilled writer. That is a rare thing.
- hodgesrm 10y agoHas philosophy ever clarified mathematics? Let's frame this question properly for analysis. All work of Aristotle is philosophy. Aristotle's work includes syllogistic logic. Syllogistic logic clarifies mathematical inference. Mathematical inference is part of mathematics. Therefore, some philosophy clarifies mathematics. QED
- joe563323 10y agoNever understood the definition of philosophy. Can philosophy be defined mathematically ?
- H4CK3RM4N 10y agoI think it's worth making it clear that Plato's contributions to political science, maths, physics, and religious thought were all framed at the time as philosophy.
- Chirono 10y agoIt strikes me that this is similar to the 'no true ai' paradox, where as soon as a program can do something, such as beat a human at chess, then it stops being AI and starts being just computation. As soon as something philosophical becomes rigerous or well understood, we stop calling it philosophy and start calling it mathematics.