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For those interested in constructive and intuitionistic approaches here Dummett's [0] Elements of Intuitionism is an extremely good read. Intuitionism is a for

by tel 10y ago

For those interested in constructive and intuitionistic approaches here Dummett's [0] Elements of Intuitionism is an extremely good read. Intuitionism is a form of a constructive foundation for mathematics which (a) notes that any attempt to deny the uncountability of reals leads to difficulties and (b) any attempt to internally define them violates constructivity. The resolution proposed is to posit the existence of "free choice sequences". These are essentially "unpredictable" sequences of potentially infinite binary choices. As there is no a priori reason to believe that they can be predicted they are able to be much larger than what is computable and thus can be used to give a characterization of the reals. Atop this you build a constructive understanding of free choice reals which behaves very nicely (at least foundationally... it points out all kinds of weirdnesses about what we assume classically to be the structure of reals). What's very nice about this solution is that it sidesteps the difficulty. Free choice is a weaker thing to ask for than finite/constructive reals, but finite/constructive reals could be transparently encoded into free choice sequences and all the math would just work. [0] https://www.amazon.com/Elements-Intuitionism-Oxford-Logic-Guides/dp/0198505248 https://www.amazon.com/Elements-Intuitionism-Oxford-Logic-Gu...

pron 10y ago
It sidesteps the difficulty by making the math itself (at least for the time being) much more difficult, and that is the reason it was rejected by most mathematicians in the Hilbert/Brouwer debates. Because here's the question: suppose you can't philosophically justify the "existence" of the real numbers, yet they coincide perfectly with observation and result in math that is much simpler than constructive math. Should you reject them? Brouwer -- who was the first to recognize just how much ordinary math relies on non-constructive principles -- said yes, because non-constructive math is philosophically wrong, period. Hilbert -- who was a finitist -- instead suggested that the propositions of math come in two flavors: real propositions, those that are finitary and can be taken to say something about physical reality, and ideal propositions, that are not. He said that as long as the ideal propositions are consistent with the ideal ones, they should not be rejected on a priori philosophical grounds even if no finitary meaning can be assigned to them. I.e. they are philosophically justified after the fact by virtue of their consistency with the real propositions. This philosophical classification of mathematics into "real" and "ideal" is called formalism, because it does not require that the ideal propositions be assigned a finitary meaning beyond their formal statement (as a finite string of characters). Of course, most mathematicians are not finitist, so they require neither intuitionism nor formalism -- both essentially finitist philosophies -- and are Platonists, believing that even ideal objects that are beyond physical reality and computation have a "real existence" in some Platonic sense. BTW, I think that after Turing (who used Brouwer's choice sequences in his construction of computable numbers) it is no longer necessary to rely on "free choice" (or lawless) sequences because of the halting theorem, and both lawlike and lawless sequences can be unified, and Brouwer's "creating subject" identified with a Turing machine. But I'm not sure about that. Turing -- who was a mathematical philosopher himself -- rejected any dogmatic a priory philosophy of mathematics, except for common sense, as the one true foundation, and suggested that the value of a formal system be derived not from its a priori philolosphy but from its ad hoc utility.
- threepipeproblm 10y ago
  Wonderful to know that Hilbert was a finitist, and that https://en.wikipedia.org/wiki/Finitism https://en.wikipedia.org/wiki/Finitism is an official camp. Without knowing too much about about the subject, I've vaguely wondered about this idea for a long time, now, but I figured it likely an un-respectable position. I think it's too bad that beginners are often shielded from controversies in foundations. Within the past few years I ran across an alternate approach to calculus, which if I recall correctly, achieves the same basic results, but without the same notion of infinitely small slices and so on... now I can't find it to link.
  - pron 10y ago
    Yep, and as you can see in that Wikipedia article, both Brouwer and Hilbert were finitists. It is not that it is an unrespectable opinion as much as mathemticians these days -- from what little I know -- are not generally expected to hold any dogmatic opinion on philosophical foundations, but maybe that attitude will shift again. > I think it's too bad that beginners are often shielded from controversies in foundations. Mathematicians generally shouldn't worry about foundations, as it is the intention of foundations to stay hidden -- except in cases where the foundation requires a rewrite of much of math, as in the case of Brouwer's intuitionism, but constructive math is pretty advanced anyway. Foundations are usually the concern of logicians, but from what little I've seen in online discussions, it seems like many logicians aren't interested in philosophy, either, and that's a real shame. > I ran across an alternate approach to calculus, which if I recall correctly, achieves the same basic results, but without the same notion of infinitely small slices and so on There's constructive analysis[1], which recreates analysis within the framework of constructive math (by finite means etc.), and there's also non-standard calculus[2] (which I know nothing about) that makes treat infinitesimals as actual numbers, which may be the opposite of what you meant, but maybe not -- I saw something about there being constructive versions of non-standard analysis, too. [1]: https://en.wikipedia.org/wiki/Constructive_analysis https://en.wikipedia.org/wiki/Constructive_analysis [2]: https://en.wikipedia.org/wiki/Non-standard_calculus https://en.wikipedia.org/wiki/Non-standard_calculus
- tel 10y ago
socmag 10y ago
All numbers are ultimately "probabilistic" in calculations. Sure you can state them as a "fact", but that undermines quantum theory itself and the Universe. If you want to do the math right you really need to consider terms as peak probabilities and work from there. (See Chaos theory) IMHO Of course doing math with hard constraints is perfectly adequate for most macroscopic calculations. That said, errors do add up given sufficient entropy and time. It's quite simple to add fuzziness into the equations but quite a pain in the ass to compute with, plus you will in most cases get the same answer anyway. As far as real numbers, they are no different from integers, just shift the decimal.
- Bigbestbooty 10y ago
  are u retrded loool

by tel 10y ago

pron 10y ago

It sidesteps the difficulty by making the math itself (at least for the time being) much more difficult, and that is the reason it was rejected by most mathematicians in the Hilbert/Brouwer debates. Because here's the question: suppose you can't philosophically justify the "existence" of the real numbers, yet they coincide perfectly with observation and result in math that is much simpler than constructive math. Should you reject them? Brouwer -- who was the first to recognize just how much ordinary math relies on non-constructive principles -- said yes, because non-constructive math is philosophically wrong, period. Hilbert -- who was a finitist -- instead suggested that the propositions of math come in two flavors: real propositions, those that are finitary and can be taken to say something about physical reality, and ideal propositions, that are not. He said that as long as the ideal propositions are consistent with the ideal ones, they should not be rejected on a priori philosophical grounds even if no finitary meaning can be assigned to them. I.e. they are philosophically justified after the fact by virtue of their consistency with the real propositions. This philosophical classification of mathematics into "real" and "ideal" is called formalism, because it does not require that the ideal propositions be assigned a finitary meaning beyond their formal statement (as a finite string of characters). Of course, most mathematicians are not finitist, so they require neither intuitionism nor formalism -- both essentially finitist philosophies -- and are Platonists, believing that even ideal objects that are beyond physical reality and computation have a "real existence" in some Platonic sense. BTW, I think that after Turing (who used Brouwer's choice sequences in his construction of computable numbers) it is no longer necessary to rely on "free choice" (or lawless) sequences because of the halting theorem, and both lawlike and lawless sequences can be unified, and Brouwer's "creating subject" identified with a Turing machine. But I'm not sure about that. Turing -- who was a mathematical philosopher himself -- rejected any dogmatic a priory philosophy of mathematics, except for common sense, as the one true foundation, and suggested that the value of a formal system be derived not from its a priori philolosphy but from its ad hoc utility.

threepipeproblm 10y ago
Wonderful to know that Hilbert was a finitist, and that https://en.wikipedia.org/wiki/Finitism https://en.wikipedia.org/wiki/Finitism is an official camp. Without knowing too much about about the subject, I've vaguely wondered about this idea for a long time, now, but I figured it likely an un-respectable position. I think it's too bad that beginners are often shielded from controversies in foundations. Within the past few years I ran across an alternate approach to calculus, which if I recall correctly, achieves the same basic results, but without the same notion of infinitely small slices and so on... now I can't find it to link.
- pron 10y ago
  Yep, and as you can see in that Wikipedia article, both Brouwer and Hilbert were finitists. It is not that it is an unrespectable opinion as much as mathemticians these days -- from what little I know -- are not generally expected to hold any dogmatic opinion on philosophical foundations, but maybe that attitude will shift again. > I think it's too bad that beginners are often shielded from controversies in foundations. Mathematicians generally shouldn't worry about foundations, as it is the intention of foundations to stay hidden -- except in cases where the foundation requires a rewrite of much of math, as in the case of Brouwer's intuitionism, but constructive math is pretty advanced anyway. Foundations are usually the concern of logicians, but from what little I've seen in online discussions, it seems like many logicians aren't interested in philosophy, either, and that's a real shame. > I ran across an alternate approach to calculus, which if I recall correctly, achieves the same basic results, but without the same notion of infinitely small slices and so on There's constructive analysis[1], which recreates analysis within the framework of constructive math (by finite means etc.), and there's also non-standard calculus[2] (which I know nothing about) that makes treat infinitesimals as actual numbers, which may be the opposite of what you meant, but maybe not -- I saw something about there being constructive versions of non-standard analysis, too. [1]: https://en.wikipedia.org/wiki/Constructive_analysis https://en.wikipedia.org/wiki/Constructive_analysis [2]: https://en.wikipedia.org/wiki/Non-standard_calculus https://en.wikipedia.org/wiki/Non-standard_calculus
tel 10y ago

socmag 10y ago

All numbers are ultimately "probabilistic" in calculations. Sure you can state them as a "fact", but that undermines quantum theory itself and the Universe. If you want to do the math right you really need to consider terms as peak probabilities and work from there. (See Chaos theory) IMHO Of course doing math with hard constraints is perfectly adequate for most macroscopic calculations. That said, errors do add up given sufficient entropy and time. It's quite simple to add fuzziness into the equations but quite a pain in the ass to compute with, plus you will in most cases get the same answer anyway. As far as real numbers, they are no different from integers, just shift the decimal.

Bigbestbooty 10y ago
are u retrded loool