# Does zero represent nothingness? kal

It might be a good idea to start with a distinction between words and things. The number zero, or 0, is one among an infinity of numbers, special though its algebraic properties may be. As such, it just is and doesn't represent anything. The numeral '0' typically represents the number 0, but doesn't usually represent nothingness, whatever exactly that is. If I write 7 - 7 = 0 I've written something true; if I write 7 - 7 = nothingness I've written something peculiar and not obviously true at all. People sometimes use the word "nothingness" in a way that's fraught with meaning - as in the Existential Void.The number 0 isn't that, and the numeral '0' doesn't normally represent it. That said, we have a very free hand in deciding what represents what. We can and do use '0' to represent all manner of things (or their absence). So whatever you have in mind by your use of the word "nothingness" we no doubt could use '0' to represent it. It's just that this would be something we decide...

# Setting aside worries about quantum mechanics, would it be possible for there to be a plank of wood which is an irrational number (say, pi) of feet in length?

Sure. For one thing, nature doesn't care about our arbitrary units. Suppose we have a plank of wood that''s exactly a foot long. Now I define a new unit: a schfoot. Anything one foot long is exactly pi schfeet long. Is there any mystery about things being pi schfeet long? Also -- since we're setting aside issues about quanta and, I assume, the possibility that space is granular, can't we make sense of something changing length continuously? A twig that's a foot long and growing will pass through an uncountable number of irrational lengths on its way from being a foot long to being two feet long.

# Why is it necessary that 2+2=4? Because it is difficult to conceive how 2+2 could have been other than 4? But how do we know that this is not just due to our limitations? The fact that we, i.e. our brains, cannot imagine a different result does not per se mean that it is logically impossible for 2+2 not to be 4 (given the standard semantics of course).

We need to keep two questions straight here: (i) why is it necessary that 2+2=4; (ii) why should we believe it's necessary that 2+2=4. The first question assumes that this is, in fact, a necessary truth, and asks what grounds the necessity. The second asks how we know. On the first: why is it necessary that 2+2=4? Part of the problem is to decide what would count as an answer. I'll leave it to wiser heads than mine to offer a thought. But I think your worry actually lies with (ii). Now of course, in some sense of "could," it could be that we're all utterly deluded, and in fact 2+2 isn't equal to 4 at all. All that this means is that I can't rule out beyond any possible doubt that we're utterly addled in ways we can't even imagine. But to quote my old colleague Dudley Shapere for the m ty- n th time, the possibility of a doubt isn't a reason for doubt. Translation: the mere fact that we can dimly imagine that we're utterly and totally confused about even the...

# Is the statement "it is wrong to torture innocent people for fun", logically necessary in the same sense as "2+4=6"? Or could there (in principle) be a universe that functions according to completely different moral laws?

I'd like to suggest a rather different take. Your question makes most sense on the assumption that there can be objective moral truths; if there can't, then no universe "functions" in accord with any moral laws. So let's assume, at least for the moment, that there are such things as objective moral truths. And now let's make a bit of a distinction. Let's agree that as things stand, it's wrong to use taser guns on babies. Could there be a universe where it was perfectly acceptable to taser a baby? If we suppose that babies are wired differently in that universe, the answer could well be yes. Perhaps the nervous systems of babies in this distant universe are set up so that applying the taser provides some sort of painless and beneficial stimulation. And so somthing that's wrong in our circumstances would be right in that far-off world, but only because some background non-moral facts differ. Now it may be that background facts about our social arrangements and our ways of understanding our own...

# Are necessary truths ultimately grounded in induction? For example truths of mathematics are said to be necessary, yet don't they make generalizations about an infinite set of numbers that are not verifiable; wouldn't this be considered induction? And if we ground our necessary truths on axioms, aren't these axioms theorems that a community has agreed to as being true and are not objectively true? Thanks for your answer, John

First, we need to set an issue aside. The word "induction" is sometimes used to refer to a certain sort of mathematical argument in which we prove something for every case by showing it for a "base" case and then showing that if it holds in the first n cases, it holds in the n+1th case. But it's pretty clear that your question is about induction as a matter of reasoning empirically from a limited set of instances to a claim about all cases, and so we'll use the word "inductive" in that way below. Here's an example of a necessary truth: every star that has planets orbiting around it is a star . Notice that it's universal; it says something about every star with planets. And if it were like "every star with planets orbiting around it is at least 3 billion years old," we could only show that it was true (assuming it is) by empirical means and thus, in a loose sense, inductively. (What we'd actually do is produce an argument from various theoretical and observational premises, but...

# Can the "real world" provide evidence that mathematical knowledge is legitimate? I think its many peoples' intuition that the successful application of math to science and engineering (e.g., that we can use math to build bridges) shows that math is true.

The question is whether what we find in the physical world could tell us whether math is true. Let's consider two sorts of cases. One is what we might call mathematical laws -- 1+1=2 is a particularly simple example. An algebraic law like x 2 - y 2 = (x+y)(x-y) is another. The second sort of case includes things like Newton's law of gravitation -- F 12 = G(m 1 m 2 )/r 2 -- or some mathematical description of the characteristics of steel beams used for bridges. This may be closer to what you have in mind. Start with the first sort of case. Suppose we have two 1-liter beakers of water. We pour them together, measure the volume and find that it's two liters. Have we confirmed the mathematical claim that 1+1=2? If so, what do we make of the fact that if we put pure alcohol rather than water in one of those beakers, when we put the two together we get about 1.94 liters? Does that count against 1+1=2? It's pretty clear that neither experiment tells us anything about whether 1+1 equals 2;...

# All throughout our educational careers, we are taught not to divide by zero. Death upon he who divides by zero. If you punch it into a calculator you get an error or undefined. But, what I want to ask is if we can display this error. In reality we can divide by certain amounts. If I have four apples, and I want two 'divide by two,' I must split the apples into even groups. I can do this for any real number. But is there a realistic model that we can divide by zero? If I get the error on a calculator, can I get that error in real life? So that this apple will simple vanish, or, God forbid, time and space unravel? I think there has to be some realistic model to divide my four apples into zero baskets.

A couple of thoughts. The first is that even though arithmetic may have been inspired by things that we do when we arrange objects like apples and baskets, arithmetic isn't "about" those concrete operations. On the contrary: suppose we "add" one rabbit to another and get 10 rabbits. Then we simply don't count what we did with the rabbits (or what the rabbits did) as the arithmetic addition operation. Likewise, if I "add" one drop of water to another, I'll get one drop. But that doesn't give us an exception to "1+1=2". Rather, we say that the sort of "adding" we do when we plop one drop on top of another isn't arithmetic adding. We could give some more or less arbitrary "operational definition" of some kind of real-world "dividing" of four apples into zero baskets, but it wouldn't tell us anything about arithmetic. There's another point. If dividing 4 by zero is going to make any sense, then the result can't be a real number (i.e., member of the set of reals). Why not? Because no matter how big a...