Intuitionistic mathematics for physics(math.andrej.com)
math.andrej.com
Intuitionistic mathematics for physics
http://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/
5 comments
One wonders whether most mathematicians and physicists reject the author's point of view because to accept it requires either (1) a deep understanding of set theory and its limitations or (2) an attitude towards truth that's rather more cavalier than you'll usually find in the faculty lounge.
I suspect that the author himself fails (1) and so must believe what he does due to (2). I will not pretend to understand the details or even the general background of all the concepts needed to get at the heart of the issue, but I will single out one example to illustrate what I mean (hopefully).
The author credits synthetic differential geometry [1] as an intuitionistic-logical basis for infinitesimals (because of its rejection of the principal of the excluded middle) and in the next sentence calls Abraham Robinson's non-standard analysis a theory based in "classical logic". Now, this must seem perfectly reasonable to anyone who has sweated through Robinson's process of constructing the hyperreals [2]. But the author seems unaware of Edward Nelson's alternative approach, called internal set theory [3], which takes an axiomatic approach.
Nelson (who, by the way, is one of those ultrafinists whose existence the author questions in his second paragraph) simply adds a few axioms to Zermelo-Fraenkel set theory and calls it a day [4]. He can do this because those axioms must be consistent if ZFC itself is consistent. Which seems unobjectionable enough, if you're not a mathematician; but if you are, you probably don't like playing with axioms. (As Bertrand Russell wrote, "The method of 'postulating' what we want has many advantages; they are the same as the advantages of theft over honest toil." He then rejects that approach, unlike Nelson, who admirably pilfers what he needs and proceeds to take a nap.)
And then you also probably don't like being reminded that ZFC's consistency (or not) is unprovable [5]. You see, Nelson's axiomatic construction of the hyperreals relies on the very same logical rabbit-hole as the author's intuitionistic logic: some things are neither provable nor disprovable in mathematics (or logic generally).
And that's why I say accepting the author's point of view requires either (1), if you realize that math is on shaky ground from the start, or (2), if don't much care and just want to get to the answer as quickly as possible (which is why I love Nelson's approach to infintesimals).
1. http://en.wikipedia.org/wiki/Synthetic_differential_geometry
2. http://mathforum.org/dr.math/faq/analysis_hyperreals.html#co...
3. http://en.wikipedia.org/wiki/Internal_set_theory
4. See Edward Nelson, Radically Elementary Probability theory, available at http://www.math.princeton.edu/~nelson/books/rept.pdf [PDF]
5. Yes, this is Gödel's first theorem: http://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theore...
I suspect that the author himself fails (1) and so must believe what he does due to (2). I will not pretend to understand the details or even the general background of all the concepts needed to get at the heart of the issue, but I will single out one example to illustrate what I mean (hopefully).
The author credits synthetic differential geometry [1] as an intuitionistic-logical basis for infinitesimals (because of its rejection of the principal of the excluded middle) and in the next sentence calls Abraham Robinson's non-standard analysis a theory based in "classical logic". Now, this must seem perfectly reasonable to anyone who has sweated through Robinson's process of constructing the hyperreals [2]. But the author seems unaware of Edward Nelson's alternative approach, called internal set theory [3], which takes an axiomatic approach.
Nelson (who, by the way, is one of those ultrafinists whose existence the author questions in his second paragraph) simply adds a few axioms to Zermelo-Fraenkel set theory and calls it a day [4]. He can do this because those axioms must be consistent if ZFC itself is consistent. Which seems unobjectionable enough, if you're not a mathematician; but if you are, you probably don't like playing with axioms. (As Bertrand Russell wrote, "The method of 'postulating' what we want has many advantages; they are the same as the advantages of theft over honest toil." He then rejects that approach, unlike Nelson, who admirably pilfers what he needs and proceeds to take a nap.)
And then you also probably don't like being reminded that ZFC's consistency (or not) is unprovable [5]. You see, Nelson's axiomatic construction of the hyperreals relies on the very same logical rabbit-hole as the author's intuitionistic logic: some things are neither provable nor disprovable in mathematics (or logic generally).
And that's why I say accepting the author's point of view requires either (1), if you realize that math is on shaky ground from the start, or (2), if don't much care and just want to get to the answer as quickly as possible (which is why I love Nelson's approach to infintesimals).
1. http://en.wikipedia.org/wiki/Synthetic_differential_geometry
2. http://mathforum.org/dr.math/faq/analysis_hyperreals.html#co...
3. http://en.wikipedia.org/wiki/Internal_set_theory
4. See Edward Nelson, Radically Elementary Probability theory, available at http://www.math.princeton.edu/~nelson/books/rept.pdf [PDF]
5. Yes, this is Gödel's first theorem: http://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theore...
The condescending tone towards Andrej Bauer is unwarranted, who is from the Dana Scott PhD stable and who has a well-thought out attitude towards constructivism and realist foundations of mathematics.
Nelson is respected but not what I would call very influential. I took this to be the point of Bauer's jibe about ultrafinistists. Note that, AFAICS, Nelson does not call himself an ultrafinitist, but rather a (radical) predicativist.
I don't get the point about the cavalier attitude to truth: constructivist rejection of PEM comes from worries that some propositions are not sharp enough to be determinately true or false, the classical Brouwerian examples coming from topology. This stance more deserves to be called pernickety than cavalier.
Nelson is respected but not what I would call very influential. I took this to be the point of Bauer's jibe about ultrafinistists. Note that, AFAICS, Nelson does not call himself an ultrafinitist, but rather a (radical) predicativist.
I don't get the point about the cavalier attitude to truth: constructivist rejection of PEM comes from worries that some propositions are not sharp enough to be determinately true or false, the classical Brouwerian examples coming from topology. This stance more deserves to be called pernickety than cavalier.
In my defense, that's a written-at-five-in-the-morning tone, not a condescending tone. I'm not sure what I was trying to write with the "I imagine the author himself fails (1)" bit, as I am sure Bauer is far beyond me on the subject, as I admitted in my comment. Apologies to Bauer and to all for coming off wrong.
Writing 'cavalier attitude to truth' was a poor choice of words, but let me explain what I meant. I agree that intuitionistic math is logically more rigorous, and it sets a higher bar for what can be considered true. But because it rejects the validity of tools like proof by contradiction and the law of the excluded middle—tools that classical mathematicians use to disprove many concepts—intuitionistic math contains a much wider universe of possible mathematical objects. It sets a lower bar for what should be considered untrue.
So when you use mathematical objects that only potentially exist—that is, objects whose existence you can neither prove nor disprove—it seems to most classical mathematicians as more than a bit dishonest, because they work only with objects that must exist. (I brought up Nelson's axiomatic approach to infinitesimals because it is analogous in this regard.) Thus Russell's likening postulates to theft.
Intutionists get to play with toys that other mathematicians don't because the former are, in a sense, more broad-minded about what toys are (could be) out there. That's all I meant by 'cavalier attitude to truth'.
Like I said, I admire this approach, and have tried to pick up some axiomatic approaches myself when they're helpful, but I still do think that Russell's bit of pith is quite apt.
Writing 'cavalier attitude to truth' was a poor choice of words, but let me explain what I meant. I agree that intuitionistic math is logically more rigorous, and it sets a higher bar for what can be considered true. But because it rejects the validity of tools like proof by contradiction and the law of the excluded middle—tools that classical mathematicians use to disprove many concepts—intuitionistic math contains a much wider universe of possible mathematical objects. It sets a lower bar for what should be considered untrue.
So when you use mathematical objects that only potentially exist—that is, objects whose existence you can neither prove nor disprove—it seems to most classical mathematicians as more than a bit dishonest, because they work only with objects that must exist. (I brought up Nelson's axiomatic approach to infinitesimals because it is analogous in this regard.) Thus Russell's likening postulates to theft.
Intutionists get to play with toys that other mathematicians don't because the former are, in a sense, more broad-minded about what toys are (could be) out there. That's all I meant by 'cavalier attitude to truth'.
Like I said, I admire this approach, and have tried to pick up some axiomatic approaches myself when they're helpful, but I still do think that Russell's bit of pith is quite apt.
OK, that is clearer.
I guess we could call the revised argument the "cavalier attitude to refutation". I don't agree about the "much wider universe": Kripke semantics draws a line around what is possible, and tells us that intuitionistic structures can be embedded in families of partial classical structures that evolve towards (but need not reach) a classical structure. Structures inconsistent with this view can be intuitionistically refuted.
I guess we could call the revised argument the "cavalier attitude to refutation". I don't agree about the "much wider universe": Kripke semantics draws a line around what is possible, and tells us that intuitionistic structures can be embedded in families of partial classical structures that evolve towards (but need not reach) a classical structure. Structures inconsistent with this view can be intuitionistically refuted.
>And then you also probably don't like being reminded that ZFC's consistency (or not) is unprovable [5].
Unprovable within ZFC. Peano arithmetic is proven consistent, but it cannot prove itself consistent. This is an important distinction, cf http://bit.ly/lGtEDm
Unprovable within ZFC. Peano arithmetic is proven consistent, but it cannot prove itself consistent. This is an important distinction, cf http://bit.ly/lGtEDm
I don't understand what you mean by cavalier, I've always though of intuitionistic logic as having a more stringent requirement for things to be true.
Thanks for mentioning Nelson's Internal Set Theory, I didn't know about it before. I will point out that the logic in IST is also classical, unlike in synthetic differential geometry were it is intuitionistic.
Thanks for mentioning Nelson's Internal Set Theory, I didn't know about it before. I will point out that the logic in IST is also classical, unlike in synthetic differential geometry were it is intuitionistic.
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Does anyone else look at the word "intuitionistic" and want to scream?
That reaction is intuitive but nobody wants to be branded as grammar n*zis, I guess
How (if at all) do you do the same thing with dz's?