As 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.
* 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.
Furthermore, 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.
Care to elaborate? After all this was exactly the subject of a conversation between Einstein and Heisenberg that the latter recounted in his mémoire "The Part and the Whole".
It might not be our modern view but it seems Heisenbergs view on the origin of QM is very much that. He wanted to work with the observable transitions directly rather than any assumed underlying reality, hence his matrix mechanics.
Quoting from wikipedia:
> Matrix mechanics, on the other hand, came from the Bohr school, which was concerned with discrete energy states and quantum jumps. Bohr's followers did not appreciate physical models which pictured electrons as waves, or as anything at all. They preferred to focus on the quantities which were directly connected to experiments.
yes, 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.
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 :) ).
In 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.
I 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.
Isn'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.
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.
The 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?
I'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)
Readit News
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.
Wow, that is such a non-sequitur that it casts doubt on the whole thing.
It might not be our modern view but it seems Heisenbergs view on the origin of QM is very much that. He wanted to work with the observable transitions directly rather than any assumed underlying reality, hence his matrix mechanics.
Quoting from wikipedia:
> Matrix mechanics, on the other hand, came from the Bohr school, which was concerned with discrete energy states and quantum jumps. Bohr's followers did not appreciate physical models which pictured electrons as waves, or as anything at all. They preferred to focus on the quantities which were directly connected to experiments.
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 :) ).
https://en.m.wikipedia.org/wiki/Mike_Alder#Newton.27s_flamin...
But only by doing philosophy!
[1] https://ncatlab.org/nlab/show/William%20Lawvere#RelationToPh...
[2] http://philosophy.stackexchange.com/questions/9768/have-prof...
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.
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
[2]: 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 published in The Universal Turing Machine A Half Century Survey, 1995
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.
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.
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
It bears some truth :)
https://news.ycombinator.com/item?id=13610134