Please list both the problem/area and justify why it is important philosophically. This question doesn't cover questions that are only important within the philosophy of mathematics itself.
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1What sorts of philosophical implications do you have in mind - philosophy of mathematics, or more general philosophy? – Carl Mummert Oct 05 '10 at 11:17
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Do you mean e.g. Philosophy of mathematics? http://en.wikipedia.org/wiki/Philosophy_of_mathematics "constructivism involves the regulative principle that only mathematical entities which can be explicitly constructed in a certain sense should be admitted to mathematical discourse. (...) some adherents of these schools reject non-constructive proofs, such as a proof by contradiction." – Américo Tavares Oct 05 '10 at 11:23
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@Carl, @Americo: I decided to exclude the philosophy of mathematics itself as it would probably be better dealt with in a separate question – Casebash Oct 05 '10 at 11:34
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-1: too open ended. there is no indication about the questioner's motivation, specific interests or of effort put on their part. – Jyotirmoy Bhattacharya Oct 05 '10 at 11:57
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I don't think this is open ended. I read it as asking for facts about what philosophical implications of mathematical results have been studied, outside of philosophy of mathematics. – Seamus Oct 05 '10 at 12:35
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None. Mathematical pursuits and results have been inspirational to all sort of things (Lacan comes to mind...), but I am more or less confident in saying that none of that counts as an implication. – Mariano Suárez-Álvarez Oct 05 '10 at 12:57
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2I show you an empty box. I then place two marbles in the box. Now I take one out of the box. Arithmetic implies something about how many marbles are in the box, on some level. It doesn't cause there to be one marble there (unless you have a wacky view on causation) but arithmetic can here be the basis for knowledge about the world. Much of physics works in a similar, if more complicated, way. – Seamus Oct 05 '10 at 13:18
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Arithmetic does not imply anything about the real world. The fact that marbles do not come out of thin air to replace the one you took out of the box to screw your arithmetic is non-arithmetical. – Mariano Suárez-Álvarez Oct 05 '10 at 13:23
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I think the OP and I are using "imply" in a much looser sense than you are. And of course, the inference I mentioned above about the marbles is relative to background assumptions about the impossibility of spontaneous generation of marbles, but it is no less a reasonable inference because of it. – Seamus Oct 05 '10 at 14:08
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If "imply" is supposed to be understood as "suggest", "inspire" or something else, then it would be good to use the correct word... – Mariano Suárez-Álvarez Oct 05 '10 at 14:17
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2"implication" is perfectly legitimate here. "President X's change of heart on Issue Y has implications for Project Z" is an acceptable use of "implication", despite there being no strictly deductive consequences involved. You're interpreting the word too narrowly. – Seamus Oct 05 '10 at 14:50
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5I won't vote to close, but I will say that in my opinion a SE-style Q&A site is designed to answer much more specific questions than this one, to which whole books are devoted to answering. – Pete L. Clark Oct 05 '10 at 22:01
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@Peter: Book recommendations are welcome, though I don't know how long it will take me to look at them – Casebash Oct 06 '10 at 00:24
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Wikipedia has a more detailed description for each one of them, therefore I will just list them and the main ideas.
Two people under certain prior conditions can not honestly disagree forever. In fact, Scott Aaronson have proved they don't have to exchange too much information to lead to an agreement. If the prior conditions are met and the disagreement lasts too long, then one side has to be dishonest.
In short, there is no perfect voting system.
Under certain assumptions, if we have free will, so does elementary particles.
Gödel's incompleteness theorems
There are statements in a sufficiently strong formal system that can't be proven true or false within the system. Some people use this to justify humans must be different from machines, since humans can prove theorems by using another formal system.
Tarski's undefinability theorem
Similar to the theorem above, it states truth in a sufficiently strong formal system can't be defined by that formal system. For people who believe people are machines, this implies people can't define truth.
The following theorems might be a stretch, but it looks like someone can use them in philosophical arguments.
It shows there is no distributed system such that each machine store the same information, can operate while some machines are broken, and can operate even when some messages are lost.
There is no algorithm to check if an infinite set have some non-trivial property.
Shannon's source coding theorem
The theorem states there is a hard bound on data compression.
Chao Xu
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+1 for "The following theorems might be a stretch, but it looks like someone can use them in philosophical arguments." – Arkady Feb 22 '12 at 23:14
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Goedel's incompleteness theorem is used by some (e.g. Roger Penrose) as part of a justification for why computers will never achieve consciousness.
The fact that all infinite dimensional separable Hilbert spaces are isomorphic has philosophical implications for the metaphysics of quantum mechanics.
Various results in dynamical systems theory related to chaotic systems limit what can be said about predictability and about what it means for a system to be deterministic. For example this paper by Ornstein and Weiss (warning: it's huge and will take a long time to download on slow connections) has been used to suggest that the distinction between deterministic and stochastic systems is flawed.
Seamus
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There is a distinction between deterministic and stochastic systems, so whatever suggestion has been made is probably misguided. – Mariano Suárez-Álvarez Oct 05 '10 at 12:53
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The fact that all separable infinite dimensional Hilbert spaces are isomorphic may have inspired claims in the metaphysics of quantum mechanics, but it surely does not have any consequences which are not purely mathematical (as Hilbert spaces only show up in a mathematical model of the physical world, it does not even have physical consequences!) – Mariano Suárez-Álvarez Oct 05 '10 at 12:54
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I should have prefaced that sentence with something like "on the assumption that Hilbert spaces are a true (or correct, or accurate...) description of the world..." – Seamus Oct 05 '10 at 13:14
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2The Ornstein and Weiss paper shows that for any chaotic deterministic system of a certain sort, there is a stochastic system that is "alpha-congruent" to it. This alpha congruence has been interpreted (by Patrick Suppes, I think) as being roughly empirical indistinguishability. So the claim is that we can't empirically tell deterministic systems from stochastic ones. It's not an uncontroversial claim, but it is an "implication" that exists in the literature. – Seamus Oct 05 '10 at 13:16
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What is suggested to be flawed, then, is not the distinction between deterministic and stochastic systems but the distinction we make between the two, in terms of their appropriateness, when constructing models for the real world. There is a difference between coffee and tea, even though for some purposes you can use either one... – Mariano Suárez-Álvarez Oct 05 '10 at 13:28
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2I don't understand the distinction between "the distinction" and "the distinction we make". Who is doing the distinguishing in the first case? Obviously there is a formal difference between mathematical models that are deterministic and those that are stochastic. But that's not the point. – Seamus Oct 05 '10 at 14:10
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Existence and uniqueness theorems for things like differential equations (the big canonical one being for ODE's) can be thought of as a philosophical foundation for a weak type of determinism.
Ryan Budney
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Game theory and fair division have implications of sorts for moral (and political) theory. The foundations of mathematics cut to the heart of logical possibility. Noam Chomsky sparked research into formal grammar, and while I don't have much knowledge of this area I believe it holds promising theory for work in the philosophy of language. The models of biological neural networks could hold implications for the philosophy of mind and consciousness. Computability theory is suggestive towards metaphysics (see e.g. Church-Turing thesis) and the philosophy of mind.
anon
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Mathematical Realism is philosophically relevant and (imo) important.
If mathematical realism is plausible (and true) then many things about the natural reality can be infered by mathematical reasoning.
Conversely if mathematical realism holds many open mathematical problems might be resolved in un-expected ways and provide technological progress in important areas of life (or death, of course it is up to the use)
Nikos M.
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