Mathematics and PhilosophyDate: 05/15/99 at 13:52:43 From: Wim Schreurs Subject: Mathematics and truth Excuse my English, please. 1. I think mathematics made a mistake. 2. I think that mathematics cannot give an exact representation of reality in the sense of perceptibility - even things that I think myself are perceptible because I can 'see' or 'hear' the things I think, because if I don't, I don't think. 3. So there is no reality beyond perceptibility. There is only perceptibility. 4. You cannot say 2 + 3 = 5, because 2 is not three is not five. How can two things (2 and 3), neither of which is identical to 5, be identical to five if they are united? 5 in itself is also an independent "being." If it weren't, it indeed could exist as a collection of at least two other 'entities'. 5. Conclusion: only 5 = 5 or 2 = 2 is a correct mathematical exercise. ALL the other mathematic "exercises" are wrong. Something can only be identical to itself. 6. Question (although I know the answer): Is this true or not? 7. Where do they examine this terrible problem further? I'm afraid of this theory. Help me please. Wim Schreurs Date: 05/17/99 at 16:55:04 From: Doctor Peterson Subject: Re: Mathematics and truth Hello, Wim. You have some deep thoughts here, and I'll try to help you think through them. Mathematics deals not with reality, but with an abstraction of reality: a "model" of just one aspect of the reality we use it to describe. For example, a number such as 2 doesn't represent any particular pair of things, but the idea of "two-ness." Man has found through long experience that things can be counted, and that the resulting numbers accurately describe one aspect of reality: if I counted two apples yesterday, and nothing has been done to them, then when I count them again there will still be two. The number doesn't tell us their color, or how they taste, or who owns them; but it describes something about them that is true of any pair of apples. It is as much something I perceive about them as is their color or taste; but when I talk about numbers, I am abstracting one property from the rest, thinking only of the "two-ness" and ignoring the "apple-ness." Now if I have the two apples I counted yesterday, and you give me three more apples, I can count them again and I know there will be five. I don't need to know anything else about the apples to know this; addition is a property of the numbers themselves, independent of any other properties the apples may have. This means I can forget about the apples themselves, and even, if I wish, forget about the process of counting and the perception involved in doing that. Here we have entered the realm of mathematics, where we deal with abstract numbers rather than specific counted items. And within this realm, since we are no longer dealing with perception, what we say can be exactly true - although when we take our conclusions back to the "real" world, we will have to deal with the possibility that our perceptions are inaccurate: we counted wrong, apples can disappear spontaneously, or whatever. I might try to apply addition to something for which it doesn't work (adding, say, a liter of sugar to a liter of water, and expecting 2 liters of sugar water); then the problem is not in the math but in the application. It is the job of science (or merely of experience) to determine what mathematical models apply to a given situation. Returning to the mathematical realm, when I say that 2+3 = 5, I am not saying that any particular 2 and 3 items are the same as any particular 5 items. I am saying, in an abbreviated form, that whenever I put together 2 of something and 3 more of them, I will have 5 in all. Also, I am not somehow "combining" a "2" and a "3" to make something new that is identical to "5"; I am performing an operation on two numbers that is the abstract representation of putting together two sets of things, in order to determine the number of things in the combined set. If you prefer, you can say I am simply stating a relation among the numbers 2, 3, and 5. "Equal" does not mean "identical"; it means that they are two different ways to describe the same reality: I can view the apples as a set of two and a set of three, or as one set of five. If I walk forward two steps, wait a minute, then walk forward three steps, I will be in the same place as if I had walked five steps all at once. So "2+3" and "5" are "equal" in the sense that they have the same meaning, or produce the same result, even though they are different symbols. You could certainly invent your own meaning for mathematical terms, so that "equals" could only be used between two of the same number (2 = 2). But that would be worthless, since it would not allow any calculations to be done; and it would neglect millennia of human experience that calculations can in fact be done. You could also say that numbers have no reality in themselves, and that is in a sense true. What makes math "work" is that numbers, which are not a part of the "real" world, nevertheless describe it very accurately. It doesn't really matter whether numbers are "real" or whether they are an "exact representation" of reality (they are correct, but not complete); they are simply a language that describes one aspect of the reality we perceive. Now, if you are concerned that there may be no reality beyond your own perception, I think you can relax. The kind of philosophy you are struggling with has trouble dealing with the surprising "reality" of math, among other things. The fact that this mathematical "language," which fits together so beautifully, turns out to fit the reality we perceive, should encourage us to believe that what we perceive has an order to it far deeper than our own imagination could build (do your dreams have such order and consistency?), and therefore has some reality beyond ourselves. Most of us, because we don't think deeply, aren't surprised that math "works"; you have recognized that there is no reason to assume that the "reality" of math and the "reality" of perception would fit together at all, so math becomes a surprise that makes you re-evaluate your philosophy. That's good. Please feel free to write back if you need more help thinking about this. Deep thinking like this can be scary, and it's good not to do it alone. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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