## Mathematics seems to accept the concept of zero but not the concept of infinity

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 place to spell them out. But examples like this don't show that mathematics rejects the concept of infinity.

As for physics, consider a very basic physical concept: an inertial system. The net force on an inertial system is 0. Physics assumes that the concept of an inertial system is a perfectly good one and indeed, a deeply important one. Whether any actual physical systems literally experience a net force of zero is another question; it depends on how things are actually arranged in the world. But physics doesn't rule it out a priori.

### 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?

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 number 1.) The integer that goes with the natural number 2 is the set of pairs

{1,3), (2,4), (3,5), (4,6)...}

So far, all these integers are positive; notice that the second natural number in the pair is bigger than the first. The integer 0 is the set of pairs

{1,1), (2,2), (3,3), (4,4)...}

What about negative numbers? They're the pairs in which the first natural number is bigger. The integer -1 is the set

{(2,1}, 3,2), (4,3)...}

The integer -2 is

{(3,1), (4,2), (5,3)...}

and so on. There's more to the story than I've presented; to present the full story we'd need to talk about addition how the definition of addition leads naturally to treating the set

{1,1), (2,2), (3,3), (4,4)...}

as 0, and how this is intimately related to treating {(2,1}, 3,2), (4,3)...} as -1, etc. The Wikipedia article at http://en.wikipedia.org/wiki/Negative_number does a good job covering the basics.

And so the positive numbers are the "backbone: of the construction of the integers.

We could go on to define rational numbers (1/2, 3,4, -5/17, etc.) as sets of pairs of integers, with in turn are sets of pairs of natural numbers. We could then define real numbers as infinite sequences of rational numbers. So in an important mathematical sense, the positive numbers (the natural numbers) are more basic.

Are the natural numbers "really" more basic in some deep metaphysical sense? I'll confess that I don't know for sure what this question means or what would count as a good answer. But other panelists who are better-versed in philosophy of mathematics may be able to say something worthwhile on that question.

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

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.

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

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 enter the square without passing thought its walls; it enters from above or below. In a 4-dimensional space, if we picked a walled cube in a 3-dimensional "hyperplane" of the 4-space, an object could enter that cube without passing though its walls: from a region outside the plane. Those with better geometric skillz than I could offer more elaborate examples. (See also Edwin Abbot's Flatland for a delightful 19th-century exploration of this theme.)

One other thought: though we can in some sense "imagine" three dimensions, when you start trying to think about the details even of three-dimensional geometry, it's interesting just how fast imagination runs out and you're forced to adopt a more purely geometrical or mathematical approach. The limits of what we can picture or grasp intuitively aren't a very good guide to what's really possible, let alone true!

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.

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. Furthermore, we can use mathematics to do science, and to make exquisitely successful predictions that we find born out in the real world. Using math for this purpose inevitably includes using negative numbers. When we do that, or when we math for more ordinary purposes, are we really somehow committing ourselves to the weird metaphysics of something beyond (or below) existence and non-existence?

It's hard to see why. Think about my net worth. Suppose I have \$500 in the bank, that the goods I owe could be sold for \$1000, and that I owe \$2000 to my credit card company. Then my net worth is -\$500. That doesn't mean that there's some kind of metaphysically shady stuff called negative money. It just means that, for example, if I cleaned out my bank account, sold my goods and gave it all to the credit card company, I'd still owe them \$500. The first point is that there's nothing mysterious there. The second point if that this is pretty much all we mean when we say that someone's net worth is a negative number. And the third point is that being able to use negative numbers for this sort of bookkeeping is extraordinarily useful. We could avoid the word "negative." We could say that I owe more than I own. But using negative numbers in this context is precisely a way of keeping track of the "owed" versus "owned" distinction.

Or think about your car example. Suppose I drive 100 miles due north, then turn around and drive 200 miles due south. Now 100 miles - 200 miles is -100 miles. You point out quite correctly that all this is really doing is telling me where I am along an axis. In this case, I'm at a point 100 miles south of my original starting point. That's absolutely right. And using negative numbers is an extremely useful way to keep track of that. The negative numbers don't carry a metaphysical commitment. Indeed, we can label points north of my starting point with negative numbers, or we could use the negative numbers for points south.

This suggests that what's really at stake here is a worry over a word. If we take the word "negative" in a certain way, it seems to point to metaphysical bogeymen. But when we look at how so-called negative numbers are actually used, we see that they don't carry any such implication.

A bit more abstractly, a mathematician might say this. We can add numbers. There's a special number, called "zero" in English, and usually written "0," that has a special property: when we add it to another number, nothing changes. And when we consider the full number system, each of the non-zero numbers has what's called an additive inverse. When we perform the operation we call addition on a number and its additive inverse, the result is the special number called 0. When I subtract the number p from the number q, I am adding the additive inverse of the number p to the number q. All of this is consistent, useful, and apart from worries about what numbers are in general (worries that aren't the same as the one you're raising), metaphysically innocuous.

So to sum up: you're quite right that we don't want to commit ourselves to fantastical states of sub-existence. But when we look at mathematics both as an internal, abstract enterprise and as it functions when we apply it, we see that talk of negative numbers doesn't bring the fantastical implications with it.

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

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...

I'm not familiar, either, with any working mathematicians who think there is a largest natural number or, more specifically, that there are only finitely many numbers. I do know of some work, by Graham Priest, that investigates finite models of arithmetical theories, but this is in the context of so-called paraconsistent logics. In Priest's theories, it is true that there is a greatest natural number, but it is also true that there isn't one! But that is probably not the kind of thing the questioner meant.

Part of the reason mathematicians are happy with infinity is that infinity is very cheap. Consider, for example, ordered pairs. If you think (a) that, given any two objects, there is an ordered pair of them and (b) that there is an object that is not a pair, then it follows that there are infinitely many pairs. Or consider the English sentences. Not just the ones someone has uttered or written down, since there are ever so many English sentences no one happens to have uttered before (such, I am sure, as the one I am currently writing), but all the English sentences there are. Is there a longest one? Surely not. Is there a longest computer program? Surely not. And so forth.

None of that is to say that there aren't large philosophical questions in this area, such as how exactly we are supposed to know that there is no largest number. There are. And, unsurprisingly, philosophers have disagreed about the answers to such questions. But it is hard to imagine doing any kind of mathematics without being able to assume that every number has a successor, or something more or less equivalent.

It's important to distinguish two different issues: (i) whether there are infinitely many natural numbers; (ii) whether there are mathematical objects that are themselves infinite. The natural numbers are, in some intuitive sense, finite objects. But the set of all natural numbers is, in some sense, an infinite object. And it is possible to accept that there are infinitely many natural numbers without accepting that there is a set of all of them or, more generally, that there are any objects that are, in their own right, infinite. And there are respected mathematicians who hold this kind of view, though they are definitely a minority.

It's also worth distinguishing the question whether there are infinitely many numbers from the question whether every number has a successor. There are natural, and important, theories of arithmetic in which one cannot prove that every natural number has a successor, but in which there are infinitely many such numbers. I explore some of these theories in my paper "Frege Arithmetic and `Everyday Mathematics'", which you can find on my website.

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

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 case of things like 1 =1) understanding what we've been told.

This sort of reply is likely to prompt someone to ask "But how do you know that you know all those things?" That question will make some people fret, but here's a perfectly good answer: I don't know how I know all those things. Coming up with a good theory of knowledge is hard work and tends to produce controversial answers. But knowing things doesn't call for a theory of how we know things. People knew things for centuries before anyone got around to asking what exactly knowledge is and how it works.

A few things do seem clear, however. One is that not everything we know comes from syllogistic or any other sort of reasoning. Another is that we can use parts of what we know to evaluate the usefulness of other possible ways of knowing things. For example: by careful investigation, we've learned a lot about the unreliability of eyewitness testimony and memory (though we haven't learned that they're never reliable.)

But the most important thing is that there's no good reason to follow Descartes and thinking that knowledge must be based on foundations that are beyond all possible doubt. That's a premise eminently worthy of doubting, not least because it does such a lousy job of accounting for something that seems much less open to doubt: that we really do know a great deal about a great many things.

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...

## Most of our modern conceptions of math defined in terms of a universe in which

Most of our modern conceptions of math defined in terms of a universe in which there are only three dimensions. In some advanced math classes, I have learned to generalize my math skills to any number of variables- which means more dimensions. Still, let's assume that some alternate theory of the universe, such as string theory is true. Does any of our math still hold true? How would our math need to be altered to match the true physics of the universe?

Let's start with a quick comment about string theory. My knowledge is only journalistic, but it's clear that string theory is a mathematical theory and states its hypotheses about extra dimensions using mathematics. And as your comment about additional variables already suggests, there's nothing mathematically esoteric about higher dimensions. When variables have the right sort of independence, they represent distinct mathematical dimensions in a mathematical space, though not necessarily a physical space. (Quantum theory uses abstract spaces called Hilbert spaces that can have infinitely many dimensions. But these mathematical spaces don't represent space as we usually think of it.)

Of course, it might be that getting the right physics will call for the development of new branches of math. Remember, for example, that Newtonian physics called for the invention of Calculus, and though earlier thinkers had insights that helped pave the way, Calculus was something new. Just what sort of new mathematical ideas science might lead to is something we'll have to wait to see. But you've raised another question: if sound physics calls for new math, would the math we have now "still hold true" as you put it?

An example might help. General relativity tells us that the geometry of space-time isn't Euclid's geometry. It's something more complicated called pseudo-Reimannian geometry. Does that mean that Euclidean geometry isn't true?

A good answer calls for making a distinction. As a mathematical construction, there's nothing wrong with Euclidean geometry and there are lots of true statements that go with it. From the axioms of Euclidean geometry, it follows that the square of the hypotenuse of a right triangle is the sum of the squares of the other two sides. Briefly, it's true that Euclidean triangles satisfy Pythagoras's rule. However, this is a statement within math itself, so to speak. Whether physical space fits Euclid's axioms isn't a mathematical question but an empirical one, and the answer turns out to be "No" (or at least "not always.")

Here's a way to look at it: math gives us ways of describing possible structures. (Euclid's axioms describe a very general sort of possible structure.) We can construct abstract proofs about those structures whether or not they fit anything in physics. A theory in science might say that one kind of structure rather than another (this geometry rather than that, this probability distribution rather than that, this kind of differential equation rather than that...) gives us the best model of some part of physical reality. But changing our mind about which mathematical structures are good models for the world doesn't amount to changing our minds about math itself.

Let's start with a quick comment about string theory. My knowledge is only journalistic, but it's clear that string theory is a mathematical theory and states its hypotheses about extra dimensions using mathematics. And as your comment about additional variables already suggests, there's nothing mathematically esoteric about higher dimensions. When variables have the right sort of independence, they represent distinct mathematical dimensions in a mathematical space, though not necessarily a physical space. (Quantum theory uses abstract spaces called Hilbert spaces that can have infinitely many dimensions. But these mathematical spaces don't represent space as we usually think of it.) Of course, it might be that getting the right physics will call for the development of new branches of math. Remember, for example, that Newtonian physics called for the invention of Calculus, and though earlier thinkers had insights that helped pave the way, Calculus was something new. Just what sort of new...

## Our professor today told us that the expression "7 + 5" is a single entity and a

Our professor today told us that the expression "7 + 5" is a single entity and a number, just like 12, and not an operation or otherwise importantly different from 12. The context was an attempt to understand Plato's aviary analogy in Theaetetus, where our professor tried to have us imagine one bird being the "7 + 5" bird and two others being the "11" and "12" birds. This seems bizarre; while 12 is obviously the result of 7 + 5, it seems that saying they are the same is like saying a cake is the same thing as its recipe. So which is it? Is a simple mathematical equation like 7 + 5 identical to its result, or is it a different kind of thing where the similarity lies only in the numeric value the two have?

Perhaps it will help to distinguish between what "7+5" refers to and how it does the referring. The expressions "7+5," "8+4,", "2x6," "36/3" and countless others all refer to the number 12. (Though not everyone agrees that there really are numbers, we'll set that issue aside here.) But they do it in different ways. Compare:

"The 44th President of the United States is Barack Obama"

This is true, and it's true because "The 42nd President of the United States" refers to the same person as "Barack Obama." Barack Obama is the same person as the 42nd President of the United States, just as the number 12 is the same number as 7+5. (Of course, the process of adding two numbers is not a number, but "7+5 = 12" doesn't say it is.)

The sense of confusion here comes from the fact that there can be more to the meaning of a referring expression than just what it refers to. The description "The 42nd President of the United States" refers to Barack Obama, as does the description "The first African American President of the United States." But the two descriptions don't have the same meaning; meaning isn't exhausted by reference.

That said, the arithmetic case raises some more complicated issues. The meaning of "The first African American President of the United States" doesn't guarantee that it refers to the same person as "The 44th President of the United States." But the meaning of "7+5" does guarantee that it refers to the same thing as "12." Just how to account for this while taking account of the fact that there is an apparent difference in meaning between the two expressions is something I'll leave to those who work on such issues. But whatever the best answer, what "7+5+ refers to is the same number that "12" refers to.

Perhaps it will help to distinguish between what "7+5" refers to and how it does the referring. The expressions "7+5," "8+4,", "2x6," "36/3" and countless others all refer to the number 12. (Though not everyone agrees that there really are numbers, we'll set that issue aside here.) But they do it in different ways. Compare: "The 44th President of the United States is Barack Obama" This is true, and it's true because "The 42nd President of the United States" refers to the same person as "Barack Obama." Barack Obama is the same person as the 42nd President of the United States, just as the number 12 is the same number as 7+5. (Of course, the process of adding two numbers is not a number, but "7+5 = 12" doesn't say it is.) The sense of confusion here comes from the fact that there can be more to the meaning of a referring expression than just what it refers to. The description "The 42nd President of the United States" refers to Barack Obama, as does the description "The first...

## Do imaginary numbers exist?

Do imaginary numbers exist?

Although the name "imaginary numbers" may suggest some special issue about existence, I think the general view would be that the existence of so-called imaginary numbers is no more and no less problematic than the existence of more familiar numbers, including zero, negative numbers and irrational numbers, all of which were considered puzzling or problematic when they first entered mathematics. Numbers, if such there be, are abstract objects of a certain sort. Whether there really such things as abstract objects at all is something that philosophers have long argued about. A bit too crudely, Platonist say yes, and nominalists say no. So if there aren't any abstract objects, then i5, for example, doesn't exist, but then neither does 5. If there are abstract objects then there's no clear reason to worry about whether 5 exists. And since the extension of the real number system to the complex number system is mathematically straightforward, there would be no clear reason to let 5 in but keep i5 out.

Although the name "imaginary numbers" may suggest some special issue about existence, I think the general view would be that the existence of so-called imaginary numbers is no more and no less problematic than the existence of more familiar numbers, including zero, negative numbers and irrational numbers, all of which were considered puzzling or problematic when they first entered mathematics. Numbers, if such there be, are abstract objects of a certain sort. Whether there really such things as abstract objects at all is something that philosophers have long argued about. A bit too crudely, Platonist say yes, and nominalists say no. So if there aren't any abstract objects, then i5, for example, doesn't exist, but then neither does 5. If there are abstract objects then there's no clear reason to worry about whether 5 exists. And since the extension of the real number system to the complex number system is mathematically straightforward, there would be no clear reason to let 5 in but keep i5 out.