Is mathematics synthetic?

Discussion in 'General Philosophy' started by Bohemian Nightmare, May 2, 2003.

  1. Bohemian Nightmare I am better than you Registered Senior Member

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    What do you think?
     
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  3. James R Just this guy, you know? Staff Member

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    Are you asking whether mathematics is discovered or created?

    If so, I'd lean towards saying it is discovered.
     
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  5. gendanken Ruler of All the Lands Valued Senior Member

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    I personally side with Frege, though somehow uneasily. He states, as many a philosopher before and since him, that math should be thought of as free floating and seperate in existence. I wouldn't have said it outright like that and it seems almost blasphemous to think so, but how else could you describe something who's initial premise you can never prove yet use it so beautifully without doing so?

    Math is incredibly beautiful.
     
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  7. blankc Your superior Registered Senior Member

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    Math is sort of common sense at it's basest level.
     
  8. proteus42 Registered Senior Member

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    What do you mean by synthetic? Do you mean, synthetic a priori ? Then the other part of the phrase is what is really worth considering: Is math a priori? Or is it a posteriori, some sort of very general natural science ?

    If you mean by "synthetic" "not analytic" and by "analytic" something like "knowable through the logical analysis of concepts" then the failure of Frege and Russell's logicist program shows that math is not analytic, it's not simply a very complicated part of logic. So, it's synthetic.
     
  9. gendanken Ruler of All the Lands Valued Senior Member

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    Says Proteus:

    "......then the failure of Frege and Russell's logicist program shows that math is not analytic, it's not simply a very complicated part of logic. So, it's synthetic."

    And you know this because you personally read what Russell had to say in his big fat volume, right?

    Say no, and you show your wisdom.
    Say yes, and I show you a liar.
     
  10. pharmakon Registered Member

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    The Russell Set and Iterative (Zermelo-Fraenkel) Set Theory

    As far as I know, the problem with both Frege's and Russell's logicist program was the discovery of the Russell Set (a set-theoretic version of the Liar's Paradox). In other words, neither of their systems could cope with "the set that contains all sets that don't contain themselves" due to its contradictory nature. Subsequent attempts to axiomatize it out of existence failed. The solution was the invention of Zermelo-Fraenkel Set Theory, whose axioms - by their very definition - don't run into the problem of the Russell Set. However, ZF only represents a system that's isomorphic to classical mathematics. Some current philosophers of mathematics propose that any language (symbolic, synthetic, or otherwise) strong enough to yield the results of classical mathematics is isomorphic. Thus they call math the science of patterns - i.e. an analytic enterprise of discovery, not creation - because it discerns patterns (that are isomorphic to some naturally, really existing structure). They call them patterns because they consider numbers to be mere placeholders in this naturally, really existing structure. Cool stuff going on in both math and the philosophy of math.
     
  11. proteus42 Registered Senior Member

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    gendanken, I was wondering for a minute whether to reply anything to this or not. But I reply because I've been asked by a learnt man who reads Platon in Old Greeks. And, of course, you probably read Thomas Aquinas in Middle Age Latin. Thought so.

    You know, gendanken, you can know that logicism failed even if you've never read Principia Mathematica. You will be surprised to hear that most people teaching philosophy of mathematics have never ever read PM from cover to cover. Have you ever seen that book? It's not for reading. It's a series of proofs. I guess it was Gödel who last took it and read it. I bet your next question will be: "Have you read Gödel's On formally undecidable propositions of Principia Mathematica and related system", please?

    I have read.

    Say no, and you show your wisdom.
    Say yes, and I show you a liar.


    You must be in a lousy mood these days, gendanken.

    Oh by the way, gendanken, you mix up several things. First, Frege's logicism, which was found faulty by Russell because of Frege's Basic Law V, and Russell's efforts to save the program that yielded ramified type theory. But it wasn't accepted by everybody either, so today we have ZF(C). But you probably know this very well... You just wanted to test me, I guess...
     
  12. spacemanspiff czar of things Registered Senior Member

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    I'm actually reading a book right now written by some cogntive scientist who are claiming that math is very much made up and not discovered. I'm in the middle of it right now, so i'm not sure how good their argument is yet.

    my gut reaction is to agree with them, but i don't really have a super great arguement for why just yet. perhaps i'll have one soon as it is my area of research.

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  13. proteus42 Registered Senior Member

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    Spacemanspiff, may I ask who are the authors of the book you're reading? It would be interesting to hear their arguments for math being made up. I've read two of Hersh's books and he wanted to prove something similar (not through cognitive science but simpler philosophical arguments) but I couldn't convince myself that he was right. I think Plato hit the truth.

    What is your area of research --- Platonism vs. constructivism in math?
     
  14. spacemanspiff czar of things Registered Senior Member

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    "Where Mathematics Comes From"
    -George Lakoff & Rafael Nunez

    It's definitly written from the perspective of a cognitive scientist trying to figure out how the mind understands mathematics. wich is good for me because that's what i'm doing.

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    I'm a cogntive scientist of sorts interested in mathematical/numerical cognition.
     
  15. Clockwood You Forgot Poland Registered Senior Member

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    I think the rule that 1+1+2 is a rule even god can not alter. (provided he exists)
     
  16. By "synthetic," do you actually mean to say "manufactured exclusively by the human mind and having no tangible or palpable root in reality" or something thereabouts?

    If not, what exactly is "synthetic" meant to mean?

    If so, I would think that mathematics is predicated solely on natural events and merely interpreted by (not contrived by) human minds, which express it in the multitudinous and often recondite forms of equations, graphs, principles, theorems, and such.

    Seeing that salient features, or crucial notions (like ab=ba, or a(b+c)=ab+ac) of the area are so indubitable and immutable, so incontrovertible and unshaking, so invariable, Mathematics could be regarded as the purest, least "synthetic," or unreal, of human concpetions.
     
  17. Xev Registered Senior Member

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    Redoubtable:

    I believe Bohemian Nightmare is referring to Kant's synthetic judgements:

    But Kant also made a less familiar distinction between analytic and synthetic judgments, according to the information conveyed as their content. Analytic judgments are those whose predicates are wholly contained in their subjects; since they add nothing to our concept of the subject, such judgments are purely explicative and can be deduced from the principle of non-contradiction. Synthetic judgments, on the other hand, are those whose predicates are wholly distinct from their subjects, to which they must be shown to relate because of some real connection external to the concepts themselves. Hence, synthetic judgments are genuinely informative but require justification by reference to some outside principle.
    http://www.philosophypages.com/hy/5f.htm

    I hope he's not, because Kant makes me want to spew.

    If he is, then, umm, based on my poor understanding of Kant, I'd say that yes, they are. A mathematical proof is self contained, yes? But it also gives us knowledge of a truth that extends beyond the mathematical proof.

    Elegently said!

    And I very much agree. I've had the notion that math is the only pure thing ever since I first saw the Pythagorean theorem.
     
  18. spacemanspiff czar of things Registered Senior Member

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    I think that alot of rules and whatnot in math only work with a set of assumptions. like the assumption in most geometry that you are dealing with a 2d flat surface. the pythagorean theorem doesn't work on a curved surface. so if one were to change these assumptions then math would change. and that is how math is synthetic or at least relative.
     
  19. wesmorris Nerd Overlord - we(s):1 of N Valued Senior Member

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    I believe that the pythagorian theorum can only be properly applied to the cartesian coodinate system. Its basis relies on the constructs from which is was conceived. As such it is obvious that the structure of further analysis would change as drastically as your new assumptions.
     
  20. Charles Fleming Registered Senior Member

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    Surely Mathematics must have been discovered! If not, then how is our manipulation of the universe, through mathematics, so successful? Lasers, Computers, satellites, all of our technology is very successful and it is all, in my opinion, because of the 'base level'[1], that is pure logic. The not, and, and or gates, are what our technology comprises of and this is 'common sense' (logic) in it's purest form, just my opinion.

    Gendanken can you give me an example, that I will be able to follow (!

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    ) that shows how 'incredibly beautiful' maths is?

    [1] Blankc: http://www.sciforums.com/newreply.php?s=&action=newreply&threadid=21737 ; 3rd reply.

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    Pharmakon you really seem to know your stuff! I have been interested but I couldn't find any books that explained the 'pure' mathematics that you write about. Could you recommend any good sources on the topic?
     
    Last edited: May 5, 2003
  21. wesmorris Nerd Overlord - we(s):1 of N Valued Senior Member

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    Hmmm.. it depends on if you think "abstract space" existed before it was "discovered" by consciousness. I think it did and continues to exist external to the human mind, however can the same interpretation be coaxed from the well of "unknown knowledge"? Will the concept of 1 arise regardless of the nature of the consciousness that searches for it? One can only guess really, but I'd guess yes.
    Have you taken a course in calculus? Oh man, it IS beautiful. Boring as HELL if you want to try to solve the problems, but man the concept involved are freakin AMAZINGLY BEAUTIFUL. It's the pinnacle of quality thinking. What is amazing is that Newton (and that other fella) could coax it from that which is unknown. Integral calculous opened a door to progress which is unparalleled. Amazing stuff.
     
  22. proteus42 Registered Senior Member

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    May I have a suggestion? Just a little test of intuitions...

    I list a few statements below, and I ask you to give me your intuitive judments about the statments. First, is it a necessary truth or a contingent one? Second, if it is judged necessary, is it a truth of logic (true in virtue of its semantic components) (= analytic) or is it like a truth of mathematics (mathematics taken very broadly) (= synthetic).


    Examples:

    "If p then q, but p, therefore q." --- 1) necessary, 2) logical (because of the meaning of "if...then...")

    "For every line L1 on the plane and point P not on L1 there is one and only one such line L2 that P is on L2 and L2 does not cross L1" --- 1) not necessary, 2) not a truth of logic (but a truth of mathematics).

    Ok, here we go:

    "All bachelors are unmarried."

    "Everything that begins will end."

    "Every cube has six sides."

    "If a number can be divided by 6, then it can be divided by 2."

    "If a part of a stick is broken off, the remaning stick will be shorter than the original."

    "Suffering is bad."

    That's all. What are your judgements?
     
  23. Charles Fleming Registered Senior Member

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    'All bachelors are unmarried.' = neccessary (in terms) and truth of logic.

    'Everything that begins will end.' = Contingent

    'Every cube has six sides.' = Contingent

    'If a number can be divided by 6, then it can be divided by 2.' = neccessary and a truth of mathematics

    'If a part of a stick is broken off, the remaning stick will be shorter than the original.' = neccessary (in terms) and a truth of logic

    'Suffering is bad.' = neccessary (in terms, again).

    How'd I do??!!

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