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Is it strange that you can't divide by zero?

It may seem strange at first blush, but there's a pretty good reason why division by 0 isn't defined: if it were, we'd get an inconsistency. You can find many discussions of this point with a bit of googling, but the idea is simple. Suppose x = y/z. Then we must have y = x*z That means that if y = 2, for example, and z = 0, we must have 2 = x*0 But if we multiply a number by 0, we get 0. That's part of what it is to be 0. So no matter what x we pick, we get x*0 = 0, not x*0 = 2. Is it still strange that we can't divide by 2? If by "strange" you mean "feels peculiar," then it's strange from at least some peoples' point of view. But this sense of "strange" isn't a very good guide to the truth. On the other hand, if by "strange" you mean "paradoxical" or something like that, it's not strange at all. On the contrary: we get paradox (or worse: outright contradiction) if we insist that division by zero is defined.

It seems to me that there are two kinds of numbers: the kind that the concept of which we can grasp by imagining a case that instantiates the concept, and the kind that we cannot imagine. For example, we can grasp the concept of 1 by imagining one object. The same goes for 2, 3, 0.5 or 0, and pretty much all the most common numbers. But there is this second kind that we cannot imagine. For example, i (square root of -1) or '532,740,029'. It seems to me that nobody can really imagine what 532,740,029 objects or i object(you see, I don't even know whether I should put 'object' or 'objects' here or not because I don't know whether i is single or plural; I don't know what i is) are like. So, Q1) if I cannot imagine a case that instantiates concepts like '532,740,029', do I really know the concept, and if so, how do I know the concept? Q2) is there a fundamental difference between numbers whose instances I can imagine and those I cannot? (I lead towards there is no difference, but I don't know how to account...

I'd suggest that while there may be differences in how easy it is for us to "picture" or "imagine" different numbers, this isn't a difference in the numbers themselves; it's a rather variable fact about us. I can mentally picture 5 things with no trouble. If I try for ten, it's harder (I have to think of five pairs of things.) If I try for 100, it's pretty hopeless, though you might be better at it than me. But I'm pretty sure that there's no interesting mathematical difference behind that. I'm also pretty sure that I understand the number 100 quite well. I don't need to be able to imagine 100 things to be able to see that 2x2x5x5 is the prime factorization of 100, for example, nor to see that 100 is a perfect square. But that may still be misleading. I have no idea offhand whether 532,740,029 is prime. But I know what it would mean for it to be prime -- or not prime. And in fact, a bit of googling for the right calculators tells me that 532,740,029 = 43 x 1621 x 7643 I can't verify that by doing the...

I have been intrigued by the theory expounded by the MIT physicist Max Tegmark that the universe is composed entirely of mathematical structure and logical pattern, and that all perceived and measured reality is that which has emerged quite naturally from the mathematics. That theory simplifies the question of why mathematics is such a powerful and necessary tool in the sciences. The theory is platonist in essence, reducing all of existence to pure mathematical forms that, perhaps, lie even beyond the realm of spacetime. Mathematics, in fact, may be eternal in that sense. The Tegmarkian scheme contains some compelling arguments. One is that atomic and subatomic particles have only mathematical properties (mass, spin, wavelength, etc). Any proton, for example, is quite interchangeable with any other. And, of course, these mathematical particles are the building blocks of the universe, so it follows that the universe is composed of mathematical structures. Another is that the vastness of the universe is...

I will confess that I don't see the charm of Tegmark's view. I quite literally find it unintelligible, and I find the "advantages" not to be advantages at all. You suggest a few possible attractions of the view. One is that "atomic and subatomic particles have only mathematical properties (mass, spin, wavelength, etc.) and hence we might as well see them as nothing but math. Any proton, for example, is quite interchangeable with any other." But first, the fact that we only have mathematical characterizations of these properties is both false and irrelevant insofar as it's true. It's false because knowing something about the mass or the spin or whatever of a particle has experimental consequences. It tells us that one thing rather than another will happen in real time in a real lab. If that weren't true, we'd have no reason to take theories that talk about these things seriously; we'd cheat ourselves of any possible evidence. Of course, we may not know what spin is "in itself," and perhaps to that...

Mathematics seems to accept the concept of zero but not the concept of infinity (only towards infinity); whereas Physics seems to accept the concept of infinity but not of nothing (only towards zero). Yet there is a discipline of 'mathematical physics' . Is there an inherent fault in mathematical physics?

I'm pretty sure that mathematicians and physicists would both reject the way you've described them. Mathematics not only accepts the concept of infinity but has a great deal to say about it. To take just one example: Cantor proved in the 19th century that not all infinite sets are of the same size. In particular, he showed that whereas the counting numbers and the rational numbers can be paired up one-for-one, there's no such pairing between the counting numbers and the full set of real numbers. Thus, he proved that in a well-defined sense, there are more real numbers than integers, even though in that same sense there are not more rational numbers than integers. Now of course, we sometimes talk about certain functions going to infinity in a certain limit. For example: as x goes to 0, 1/x goes to infinity, even though there is no value of x for which the value of 1/x is infinity. Rather, we say that at 0, the function is not defined. There are good reasons why we say that, though this isn't the...

Are positive numbers in some way more basic than negative numbers?

In more than one way, the answer is yes. It's clear that psychologically, as it were, positive numbers are more basic; we learn to count before we learn to subtract, for instances, and even when we learn to subtract, the idea of a negative number takes longer to catch onto. Also, the non-negative numbers were part of mathematics long before the full set of integers were. (In fact, treating zero as a number came later than treating 1, 2, 3... as numbers. Also, we can start with the positive numbers and define the set of all integers. The positive numbers are usually called the natural numbers in mathematics, and N is the usual symbol for the natural numbers. The integers Z are sets of ordered pairs of natural numbers on the usual definition. The integer that "goes with" the natural number 1 is the set of pairs {(1,2), (2,3), (3,4), 4,5)...} (By "goes with" I mean it's the integer that, when we're through with the construction, we can in effect, treat as the same thing as the natural...

Is it ethical for game theory to be applied to conflicts which may involve mass human deaths for non-defensive wars?

Perhaps it depends on what sort of application you have in mind. Suppose we want to understand the sorts of conflicts you've singled out. Surely the attempt to understand isn't immoral—quite the opposite given what's at stake. And suppose that the branch of mathematics known as game theory helps us come to that understanding. It's hard to see what the objection could be. On the other hand, if a country has unjustly gone to war against another country and uses game theory to come up with strategies for winning, then we might want to say that this is an immoral use of game theory. However, the immorality here has nothing special to do with game theory. What's wrong is the waging of the war in the first place.

Are dimensions exceeding 3 actually comceivable or are they purely intellectual constructs? Is this even debated in philosophy?

If I understand your question correctly, it's whether there really could be more than three dimensions in physical space. The best answer, I should think, is yes. One reason is that there are serious physical theories that assume the existence of more than three spatial dimensions: string theory is the example I have in mind. More generally, though, it's not clear why we should doubt that this is possible. The fact that we can't represent it to ourselves imaginatively doesn't seem like a very good reason. We can't represent curved space-time to ourselves imaginatively, but if general relativity is right, space-time does curve. We have a notoriously hard time representing quantum mechanical objects to ourselves imaginatively, and yet quantum mechanics is the cornerstone of much of our physics. We can even say things about what it would be like to live in a world with more than three spatial dimensions. Consider: think of a plane in 3-space, and imagine a walled square in that plane. An object can...

Hi, I love your website and I have enjoyed reading the articles. Please could you help me with a question? I would like to ask the question regarding 'negative numbers'. Can there be such a thing as a negative? Please allow me to explain. My daughter recently brought home some Math homework that asked what -20 + -10 =. So this had me thinking, -20 (or-10) does not exist. There is no difference between having no apples to having minus a million apples both equal me having no apples. I don't think this is the same as debt as the amount in question (as in financial debt) does physically exist, even if you owe it. My daughters teacher explained that you have to see it as a scale. But I do not feel this explains the question either. For example if a car travels one direction on a scale (say North) at 100mph, if the scale is reversed the car is not travelling minus 100mph, it is now simply travelling South at 100mph. Scale I feel is inaccurate, surly its a measure of direction along an axis i.e. left or right...

If I understand you correctly, there's a plausible point behind you question: things either exist or they don't. There's no such thing as what I'll call "negative existence" for shorthand, if that means a state that's somehow less than plain non-existence. And while there's no view so strange that some philosopher won't defend it, I'm betting most philosophers will agree: "negative existence" is a confused idea. I'm pretty sure mathematicians will be equally willing to go along with that. And since the point about negative existence seems so uncontroversial, that suggests we need to ask: when people use negative numbers, do they really mean to suggest that there's something "below" non-existence? I don't think so. Start with numbers themselves. There's a long-standing debate abut whether numbers of whatever sort exist, but we can sidestep that. There's a consistent, useful and highly successful enterprise called mathematics. From it, we learn all sorts of interesting and surprising things....

Hi, I was hoping for some clarification from Professor Maitzen about his comments on infinite sets (on March 7). The fact that every natural number has a successor is only true for the natural numbers so far encountered (and imagined, I suppose). Granted, I can't conceive of how it could be that we couldn't just add 1 to any natural number to get another one, but that doesn't mean it's impossible. It seems quite strange, but there are some professional mathematicians who claim that the existence of a largest natural number (probably so large that we would never come close to dealing with it) is much less strange and problematic than many of the conclusions that result from the acceptance of infinities. If we want to define natural numbers such that each natural number by definition has a successor, then mathematical induction tells us there are infinitely many of them. But mathematical induction itself only proves things given certain mathematical definitions. Whether those definitions indeed...

Prof. Maitzen will. I hope, offer his own response, but I'm a bit puzzled. First, I'm not sure which professional mathematicians you have in mind, but that's not so important. Let's start elsewhere. The usual axioms of arithmetic do, indeed, tell us that every natural number has a successor. From that it follows with no need for induction that there's no largest natural number. For suppose N is the largest natural number. Then N+1, its successor, is also a natural number, and is perforce larger than N. So I'm tempted to ask if I'm missing something. The problem I'm having is that I don't know what I'm being asked to contemplate. Perhaps there's some sense of "possible" (though I'm not convinced), on which it's possible that we're so massively deluded that we can't even get simple arguments like the one just given right. But in that case, all argumentative bets are off. Put another way, if we're wrong in thinking there's no largest natural number, then we're so hopelessly confused that...

We use logic to structure the system of mathematics. Lord Russell was described as bewildered upon learning that original premises must be accepted on some human's "say so". Since human knowledge is so fragile (it cannot have all conclusions backed up by premises), is the final justification "It works, based on axioms accepted on faith"? In short, where do you recommend that "evidence for evidence" might be found, if such exists in the anterior phases of syllogistic construction. Somewhere I have read (if I can rely upon what little recall I still have) Lord Russell, even to the end, did not desire to rely on inductive reasoning to advance knowledge, preferring to rely on deductive reasoning. Thanks. Your individual and panel contributions make our world better.

I was intrigued that you take human knowledge to be very fragile. The reason you gave was that there's no way for all conclusions to be backed by premises, which I take to be a way of saying that not all of the things we take ourselves to be know can be based on reasoning from other things we take ourselves to know- at least, not if we rule out infinite regresses and circles. But why should that fact of logic (for that's what it seems to be) amount to a reason to think that knowledge is fragile? Most of us - including most philosophers and even most epistemologists - take it for granted that we know a great deal. I know that I just ate lunch; you know that there are people who write answers to questions on askphilosophers.org. More or less all of us know that there are trees and rocks and that 1+1 = 2 and that cheap wine can give you a headache. Some of the things we know call for complicated justifications; others don't call for anything other than what we see when we open our eyes or (as in the...

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