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Philosophy Reading Group / Kant & Math

  1. Aug 22, 2009 #1
    In order to promote the informed discussion of philosophy, I thought it might help if we could all be on the same page on some of the issues, literally. I would like to invite suggestions on relatively accessible philosophy texts that we could read and discuss. What issues have been on your mind? What can I read to get up to speed so we can have an informed discussion on the topic? If you are unsure of what text is out there, feel free to just ask about an issue and I can probably suggest or find a relevant article.

    Hopefully a few people at least will get involved and we can share our thoughts and generate public discussion on the board. Obviously this would just be an unofficial and informal group.

    Although I would really rather respond to inquiries or suggestions, let me suggest a text and initial question to get the ball rolling. It can't hurt to start with a classic, so how about Kant's "The Critique of Pure Reason?" I don't want to scare anyone off, so let's start with a single page in the text - Introduction Part V.1. It's page 19-20 here: http://www.e-text.org/text/Kant%20Immanuel%20-%20The%20Critique%20of%20Pure%20Reason.pdf" [Broken]. Specifically, I'll start with a question on the quoted section below (but please read the rest of the section at least).
    We might, indeed at first suppose that the proposition 7 + 5 = 12 is a merely analytical proposition, following (according to the principle of contradiction) from the conception of a sum of seven and five. But if we regard it more narrowly, we find that our conception of the sum of seven and five contains nothing more than the uniting of both sums into one, whereby it cannot at all be cogitated what this single number is which embraces both. The conception of twelve is by no means obtained by merely cogitating the union of seven and five; and we may analyse our conception of such a possible sum as long as we will, still we shall never discover in it the notion of twelve. We must go beyond these conceptions, and have recourse to an intuition which corresponds to one of the two− our five fingers, for example...​
    Do you agree with Kant here? Do you agree that our conception of the number 12 depends to some degree on experience and isn't something we can derive rationally?

    Please feel free to jump in. No other background is necessary, although the first parts of the text could be helpful to read. Play along on this one and then suggest your own. It could be a book or a recent article or paper - anything. Let's just try to concentrate on one at a time to focus discussion and allow time for reading.
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Aug 22, 2009 #2
    And to start it off... I personally don't buy it. I'm of the belief that we don't need to learn anything from our environment to know that 12=7+5. We can figure this out completely from the definitions of the words without visualizing anything.

    Can this be proven though? The criteria he uses make it tricky. It almost turns into a problem of psychology. Are we capable of counting before we see or feel anything? I think this is the wrong direction to go when we are discussing necessary vs contingent truth. Psychology can only explain the strength and source of our belief, not its truth. Or is psychology the only resource we can look to here?
    Last edited: Aug 22, 2009
  4. Aug 22, 2009 #3
    I think math is a way of generalizing experience. Before you can get to 1+1=2, you need to understand numbers are abstractions, a second order abstraction, that is different from a physical object (first order abstraction). You can apply '1' to any object, and group objects according to categories. However, once you have a conceptual framework built up that corresponds to the world of objects, you can use logic to gain new knowledge, but only based on how accurate your framework is, at representing those real world objects.

    Even real world objects are abstractions, mainly based on their utility or lack thereof. (This is not an ontological statement about their physical nature, but rather how we relate to what is out there)
    A rock is something we can pick up, a bunch of rocks we can count, lots of rocks, which we can't be bothered to count, is 1 pile, a lot more is 1 hill.... or 1 mountain...etc..

    However, abstractions, first or second order, don't always correspond to the external world. I don't think you can get to 1+1=2, in any useful way, without real world experience for correspondence or validation. Your system could include it, but it could also include 1+1=3. Experience gives abstraction truth value.

    Fortunately for mathematicians, our current conceptual framework has been built over centuries, designed and refined to correspond to observable phenomena.

    Unfortunately, our conceptualization of real world objects is based on a limited evolved perspective. Which is why quantum and cosmic logic tend to boggle our brains. Even on the everyday level, our conceptual framework of objects can fail us, fail to correspond, to meaningfuly describe the world. And to a certain degree, since math is more abstract, but also more rigourous, math leads to fewer errors than our more mundane object framework.

    Sorry, thats not well organized. But I'm trying to keep it as simple as possible.
    Last edited: Aug 22, 2009
  5. Aug 22, 2009 #4
    I fully agree. We need to learn the language before we can 'speak' and understand it. With out knowing '1+1=2', '1+2=3', ect there is no means of deducing the answer. Even the process needs to be experienced before it can be understood.
  6. Aug 22, 2009 #5
    No this is one of the places where Kant is obviously false. He splits the world into the synthetic and analytic statements, but let me answer the math part first. Kant argues that the concept of 12 is synthetic. By now we have shown that we can derive most of mathematics from a small set of axioms. The english word "implies" shows nicely that, whatever we derive as a result from the axioms was already present. So mathematics is analytical, and every proof is a consequence derived from the axioms, we don't create new mathematics, but we discover it. We have to excuse Kant here, because he came long before Peano, and didn't really grasp the nature of numbers like we do now.
    Apart from bragging about all the things he knows about philosophy, I think there was another point to the chapter that you are reading, but I find it hard to put into words. If you take some simple pseudophilosophical statement like "Everything is love." Then Kant is annoyed by arguments that take something like this and answer unrelated questions with it. So say "Is capital punishment acceptable?" and a person says "In our great love for society it is our duty to protect it by killing those who do harm to it." What this person is doing is taking his concept of love and elaborating on what he understands love to be. But Kant says there is no new information in this. The person is just doing circles around his concept of love. Where he should really produce new knowledge with synthetic statements by connecting new things to the concept of love, not by searching for thing already contained in that concept.

    Well at least that's how I read Kant...
    Last edited by a moderator: May 4, 2017
  7. Aug 22, 2009 #6
  8. Aug 22, 2009 #7
    Hey first kudos to you for starting this. I think it's a great idea and I hope that it is successful.

    I find this view of mathematics pretty interesting. I assume with it math is no longer transcendental... or at least we can not say that the math we use is the same as the math an alien species would use.

    I think that I believe sort of a middle-road. I think mathematics isn't a human cognitive phenomena, that it exist seperate of us but we can only understand it through experiences. There's no doubt that 1+7=8 is a purely human statement however the concept is universal. That is to say that the numbers the functions etc are given names and definitions by humans but without these names and without even humans minds if one rock joins with another rock there are more than originally, this will never change.
  9. Aug 22, 2009 #8
    The problem is really Gödel. First order logic explains a lot, but once we use recursion we are troubled. Many types of strange loops will lead to valid math like the group A of all groups containing the group A, whereas the group B containing all the groups not containing B will not. Mathematics still doesn't handle this elegantly (yet) and there are still a few things out of reach but f.a.p.p. mathematics derives from our basic axioms.
  10. Aug 23, 2009 #9
    A pretty big problem from what I have read. I think you're idealizing mathematics, like the ancient greeks did with geometry. It all happens in the human brain. And different cultures have developed different mathematical systems. Ours has simply absorbed all the aspects we find most useful.

    Math is in large part analytic, but its axioms and basic logic are derived from experience. All logic comes from how we see the world working. Math is just a way to represent and predict experience using highly abstract language.

    Axioms are little more than assumptions, or constraints. And those constraints are based on our experiences in the world.
  11. Aug 23, 2009 #10
    And if you smash em together really hard you get lots of rocks.
  12. Aug 23, 2009 #11
    You my friend, have just discovered division.
  13. Aug 23, 2009 #12
    Only if one rock shatters.

    If both shatter, you have addition and division.

    And if you heat those pieces enough, 50 + 50 = 1

  14. Aug 23, 2009 #13
    I'm taking notes no worries
  15. Aug 24, 2009 #14
    Uhm... Ok.
  16. Aug 24, 2009 #15
    Omnium rerum homo mensura est.
    Well I beg to differ from you, Protagoras and all the evil Sophists. It is true, that we do not do math at random. The definition of a continuous function is motivated by our experience in the world. Nonetheless everything that was proved about it, has always been true, had we invented the concept or not. Maybe I'll borrow from Kant and call logic an a priory knowledge or go more towards Platonian ideas and say that mathematics exists irrespective of formulation. But the idea that mathematics is invented rather than discovered, is dangerous. Same goes for science as this snippet from "Surely you are joking Mr. Feynman" should illustrate:

  17. Aug 24, 2009 #16

    Fortunately I am in a time and place where I can't be burned at the stake for having evil ideas that go against doctrine. Funny thing about those sophists, one of the reasons Socrates said they were such bad people was they charged for their services, much like academics do today. Socrates would not be a fan of copyright. But some say the great teacher of Plato was really a sophist at heart, he was just fed by ego. Wisdom is knowing you know nothing, after all.

    If you can show how math exists, the stuff that it is made up of, that is separate from the human mind, then you might be different from Pato, who believed in a fantastical higher realm of reality where his forms existed. Otherwise, defining math as discovered is like defining ice cream as such; it was always there, someone just had to find it.

    And Feynman is talking about physics (science), which deals directly with descriptive empirical facts, not just with math, which is an abstraction from empirical fact. I did try and address the importance of this earlier. There is quite a history of trying to reconcile empiricism and rationalism, something that most recently the Logical Positivists tried and failed to do.

    Even so, science advances when people have crazy ideas and then make them work. Math works whether it describes reality or not, its simply more useful when it does, and its the useful math that people get charged to learn.

    But I like to live on the edge, and its been so long since I've been slandered with the title sophist, I'm quite amused, so you are in no danger from me. For the record, I don't charge, and I'm more into learning than teaching.
  18. Aug 24, 2009 #17


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    Unless you just flatly reject the idea that all knowledge is modelling, then we should be careful of applying the same words to describe what stands on either side of the epistemic gap.

    So it is dangerous to say I have mathematical ideas and reality is also mathematical. By this collapse of jargon, you are eliminating the last essential vestige of doubt that keeps us epistemologically honest.

    So why not say I have mathematical ideas and reality seems to have deep patterns? And so I can hope that, with care and testing, my mathematical ideas will come asymptotically close to capturing the "truth" of those deep patterns?

    Those patterns would of course exist irrespective of whether my formulations are very good or perhaps quite partial or faulty.

    So don't say A = A, but rather A = A'.

    One claims an identity, which leads to endless arguments about: well how do you actually know? The other claims only a mapping. And the proof then is in the pudding.
  19. Aug 26, 2009 #18
    I prepared a long answer, but my browser ate it...
    Here the short form:
    There is a problem with existence, which I tried to attack earlier by introducing a new word for the patterns you are mentioning, because I wouldn't say that they exist the same way as chairs do. I said they "besist", but it didn't find support.

    The main conflict which I deem to be irresolvable is this:
    You can argue that logic works the way it does, because it is how our brains work. So logic depends on the mathematical patterns of physics that the universe displays. But on the other hand we are minds, our discussion is based on logic (mostly :wink:), so we can argue that logic needs to be established before we analyze nature, because our minds run on it.

    Well let me correct this, it's what my mind runs on, whereas you are just a peculiar physical phenomenon, to talk to on a web forum :tongue2:
  20. Aug 26, 2009 #19
    If i recall the reason you didn't find support is because you want to invent a word to describe something we already have words for... whats the use?

    As well what do you mean by this whole 'we are minds' stuff... I don't follow.
  21. Aug 26, 2009 #20


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    If two things are inextricably linked - like models and worlds - then you can't get one right ahead of the other. Instead it is all about the working relationships that develop.

    Brains and realities have developed a relationship spanning circa 600 million years. So they have a wired in logic - which is dichotomistic. Figure-ground, attention-habit, etc.

    Humans with language invented a more partial (if locally penetrating) logic in syllogistic reasoning and other analytic methods. This is not wired into the brain but a socially constructed habit. It is still "logic", but a subset of what we actually use - and what the universe also "uses".
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