# 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...

# 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...

# 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...

# 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.

# Is there a correct formulation of set theory? For example, it's been proven by Gödel and Paul Cohen that the continuum hypothesis can neither be proven nor disproven in ZFC. Should we take from this that there exists a hitherto undiscovered formulation of set theory that can conclusively establish whether the continuum hypothesis is true or false?

Logically wiser heads (such as Peter Smith) may want to weigh in, but I think we can at least say this much. As you note, the axioms of Zermelo-Frankel set theory don't answer the question. We can add either the Axiom of Choice or its denial and to the basic ZFC axioms. As long as the basic axioms are consistent, the result will be consistent too. The same goes for the Continuum Hypothesis. But by itself this doesn't give us any reason one way or another to think that there's some more agreeable formulation without this ambiguity. And even if we came up with a particularly elegant formulation of set theory that didn't leave these axioms dangling, it's not clear that this would settle the question of truth conclusively. Behind this lies a harder question: what would it mean in the first place for the matter to be settled conclusively? The problem is that just what truth amounts to here is not a settled matter, and not a matter that formulations of axioms by themselves can settle.

# 5+5=9 is not an empirical fact. However it can be proven empirically (put 5 objects and four objects together, then count the result). How is it possible for non-empirical facts to be proven empirically?

Counting things is something we actually do to get to the answers to arithmetic problems (who hasn't counted on their fingers at some point?) but we need to be careful. What if you put five drops of water together with five drops of water and count the results? Or what if you put an electron and a positron together? We might say that putting drops of water together or combining electrons and positrons doesn't count as addition. But that's because when when we do, we don't end up with the right answer to the arithmetic question we're supposedly trying to settle empirically. What we see is that arithmetic isn't an hypothesis about what we'll find when we count in various cases. Rather, arithmetic tells us whether an empirical operation is a good model of addition, or whether what we're mumbling under our breath as we flex those fingers really counts as counting.

# What is the relationship between mathematics and logic?

It's a good idea to start with a distinction. If by "logic" you simply mean something like "correct deductive reasoning," then logic is something mathematicians use -- as do people in any discipline. If by "logic" you mean the study of certain specific kinds of formal systems and their properties -- mathematical logic, as it's often called -- then logic is arguably a branch of mathematics, but also of philosophy (and perhaps also of other disciplines such as computer science; no need for turf wars.) There are people in math departments who specialize in logic, and also people in philosophy departments. Results in mathematical logic, might be published in math journals or in philosophy journals or in computer science journals.