# Any two sets have different conditions for membership, so if object O is in set S because it's blue or green, then being blue or green cannot be the reason why any other object is in any other set. If so, how can there exist a set of green objects?

### Suppose S is the set of all

Suppose S is the set of all things that are blue or green. Then my mug is in S because it's green and therefore satisfies "x is blue or x is green," and my pen is in the set S because it's blue and therefore satisfies "x is blue or x is green." Now it's true: satisfying "x is blue or x is green" picks out only one set: the set of all things that satisfy "x is blue or x is green." But the condition "x is green" is a different condition, and so is "x is blue." However: when you say "being blue or green cannot be the reason why any other object is in any other set," there's an ambiguity. That could be read as "being blue cannot be the reason why an object is in any other set and being green cannot be the reason why an object is in any other set." In that case, however, it's false. Being green, and hence satisfying "x is green" puts my mug in the set G of all green things, and in the set S of all things that are either green or blue—that satisfy "x is green or x is blue." These two sets are not the...

# Two different sets cannot have the same reason for membership, so if beauty is the reason why a painting is in the set of beautiful paintings, then beauty cannot be the reason why the painting is in any other set, such as the set of good paintings. Is that fair?

### No.

No. There is a set of even numbers. There is also a set of numbers that are even or prime. (Note, by the way: something can be even and prime: the number 2.) The number 8 is in the first set because it's even. It's also in the second set because it's even, hence even or prime. Not all good paintings are beautiful, but for present purposes, we can still assume that all beautiful paintings are good paintings. A beautiful painting clearly fits the membership condition for the set of beautiful paintings. But it also fits the membership condition for being in the set of paintings that are beautiful or good and it fits it by virtue of being beautiful. There's nothing peculiar here at all. If X and Y are both sets, their union is also a set. That's elementary set theory, and it's so whether or not X and Y are mutually exclusive.

# Lots of science today (meteorology, cosmology) is based on computer simulation or modeling for those phenomena that are difficult to observe directly. If a computer simulation gives me a result consistent with what we can see (star distribution for two galaxies that collide) can we infer that the underlying process is the same in the simulation and in physical world? The simulation is just numbers (or symbols) input as data about the system(s) modeled. Are numbers the underlying "stuff" of objects, too, rather than atomic particles, etc.?

### Suppose that instead of a

Suppose that instead of a computer producing a simulation, we have an army of thousands of worker-bee science grad students performing and assembling vast numbers of calculations matching all the steps that a computer simulation would call for. Suppose the results are consistent with observation. We wouldn't ask whether what's being simulated is really nothing but desperate grad students chained to desks doing tedious math. Computer simulations are ways of finding out what our equations and assumptions entail. In an example where it would be feasible to calculate the behavior of the model by hand, we wouldn't doubt that the real target of the exercise is external-world stuff—particles or economic agents or pathogens or whatever. That doesn't change if we move to cases where there's no serious possibility of doing the calculations by hand. The computer simulation doesn't represent itself . It represents what it simulates. If we've done things right, the representation will be more or less accurate....

# Consider the mathematical number Pi. It is a number that extends numerically into infinity, it has no end and has no repeating pattern to its digits. Currently we have computers that can calculate Pi out to many thousands of digits but at a certain point we reach a limit. Beyond that limit those numbers are unknown and essentially do not exist until they are observed. With that in mind, my question is this, if we could create a more powerful computer that could continue to calculate Pi beyond the current limit, and we started at exactly the same time to compute Pi out beyond the current limit on two identical computers, would we observe the computers generating the same numbers in sequence. If this is the case would that not infer that reality is deterministic in that unobserved and unknown numbers only become “real” upon being observed and that if identical numbers are generated those numbers have been, somehow, predetermined. Alternatively, if our reality was non-deterministic would that not mean that...

### You're right that any

You're no doubt right that any computers we happen to have available will only compute π to a finite number of digits, though as far as I know, there's nothing to stop a properly-designed computer from keeping up the calculation indefinitely (or until it wears out.) But you add this: "Beyond that limit those numbers are unknown and essentially do not exist until they are observed" Why is that? Let's suppose, for argument's sake, that we'll never build a computer that gets past the quadrillionth entry in the list of digits in π. Why would than mean that there's no fact of the matter about what the quadrillion-and-first digit is? What does a computer's having calculated it or (at least as puzzling) somebody having actually seen the answer have anything to do with whether there's a fact of the matter? To be a bit more concrete: the quadrillion-and-first digit in the decimal expansion of π is either 7 or it isn't. If it's 7, it's 7 whether anyone ever verifies that or not. If it's not 7, then it's...

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

### It may seem strange at first

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

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

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

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.