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When the word" exist "occurs like "numbers exist "does it mean what it means in

When the word" exist "occurs like "numbers exist "does it mean what it means in sentences like "Dogs exist"?

I think it does, or at least I think the burden of proof is on anyone who says that "exist" is systematically ambiguous, meaning one thing when applied to numbers and another thing when applied elsewhere.

It's widely held that abstract objects such as numbers, if indeed they exist, don't exist in spacetime, whereas concrete objects such dogs clearly do exist in spacetime. But that doesn't affect the meaning of "exist" itself. In particular, it doesn't imply that "exist" means "exist in spacetime." Otherwise, the expression "exist in spacetime" would be redundant and the expression "exist but not in spacetime" would be self-contradictory, neither of which is the case.

Analogy: It's a fact that some things exist aerobically and some things exist anaerobically, but that fact doesn't tempt anyone to say that one or the other kind of thing doesn't really exist, or to say that "exist" just means "exist aerobically." So I see no reason not to say that numbers, if they exist, exist nonspatiotemporally, whereas dogs exist spatiotemporally: the adverbs differ in meaning, but not the verbs.

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 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.

Is 0 and infinity the same thing or are they direct opposites?

Is 0 and infinity the same thing or are they direct opposites?

Pretty clearly, zero and infinity aren't the same thing. For example, the number of prime numbers is infinite and (therefore) definitely not zero.

But I'm not convinced that zero and infinity are opposites either. (I'd be more inclined to say that negative infinity and positive infinity are opposites.) One reason is this: "zero" and "none" are often synonymous, as in "I own zero unicorns; I own none." The opposite of "none" is "all" (whereas the contradictory of "none" is "some"). But "all" and "infinitely many" are not synonymous: for example, even if we collect all the grains of sand in the world, we will collect only finitely many grains.

Is Math Metaphysical?

Is Math Metaphysical? Math is not physical (composed of matter/energy), though all physical things seem to conform to it. Does this make Math Metaphysical and mathematicians Metaphysicians?

I have no problem at all with what Stephen says, but would add a couple of things. First, Stephen didn't address what might actually be the questioner's main concern, i.e. whether the fact that "all physical things seem to conform to it" makes mathematics metaphysical. What is "it" here? Mathematics keeps growing, and one of the main sources of growth is that new things keep coming along (such as new scientific findings) for which existing mathematics is no help. The formulation of general relativity, for instance, required new mathematics that had been developed to some degree (by Riemann and others) before 1915, but without any thought that it might someday actually apply to something in the world out there. The further development of differential geometry was largely in response to its employment in theoretical physics (though of course it then took on a life of its own, as mathematical ideas do).

And these new developments invariably (perhaps inevitably) don't quite fit, in various ways, with the existing corpus of mathematics; it takes a while for a perspective to develop from which it can be assimilated and seen to be part of the same system of thought as what was there before. There seem to be some cases, in fact, where the new stuff just doesn't fit, and then it takes a while to get to the bottom of that difference. Meanwhile, new developments in logic and mathematics keep threatening to make such sluggish lucubrations out of date.

So I would answer the question by casting doubt on the idea of a stable "it." Mathematics is not one single, stable, definable language or system with which nature turns out, post hoc, to be in accord.

Secondly, I would point out that the question raised here lies at the heart of the history of philosophy. It seems essentially to be the question that got Plato started. He, and generations after him (even Kant, to some degree) were inclined to answer "yes" to the second question asked by the questioner. But in the early 1920s, Wittgenstein figured out a compelling way to answer "no," and the Vienna Circle (and all of 20th-century scientific philosophy in their footsteps) essentially took that "no" as their starting point. While I sympathize with that viewpoint, I would certainly want to acknowledge the historical importance of this question in the development of philosophy over the centuries. It was one of those deep questions that had to be asked, and has taken a long, long time to answer.

For the philosophically unsophisticated, why is it significant that logic cannot

For the philosophically unsophisticated, why is it significant that logic cannot be reduced to mathematics? What difference would it have made if that project had succeeded; what is import that it failed?

Your ability to balance your checkbook, or to draw logical inferences in everyday life, won’t be affected in the least by difficulties in figuring out just how logic and higher mathematics are connected. Nevertheless, the relationship between logic and mathematics has been an intriguing conundrum for the better part of two centuries.

There have been many attempts to understand various aspects of logic mathematically, and perhaps the most famous is George Boole’s Mathematical Analysis of Logic (1847), which laid the foundation for Boolean algebra. Far from being a failure, Boole’s effort seems to have been a smashing success, especially when we consider the extent to which Boolean algebra underlies modern digital computing.

Nevertheless, the relationship between logic and mathematics can go in two directions, not just one, and so, just as one might try to understand various parts of logic mathematically, one can also try to understand various parts of mathematics logically. It is this further possibility, I suspect, that has prompted your question about a “failure.”

Late in the nineteenth century, the German logician Gottlob Frege sought to understand part of mathematics in terms of logic. Frege wanted to reduce arithmetic to logic, and later writers tried to reduce other parts of mathematics to logic too. Today, this approach is usually called “logicism,” and the primary motivation behind it is to discover exactly what kinds of entities mathematical objects are.

When we do arithmetic, for example, we add numbers, but what exactly is a number? Is it a physical object? Is it just an idea in our heads? Is it a mere symbol? Is it a timeless, placeless eternal entity that exists even if no one thinks about it? These questions about numbers are as old as Plato (maybe older), and they generally fall under the heading of ontology—which asks what kinds of objects exist. Logicism is an attempt to answer these ontological questions, and this is why it seeks to “reduce” mathematics to logic.

In 1931, Kurt Gödel demonstrated that no logical system rich enough to include arithmetic as a consequence could be shown within that system to be both consistent and complete. Either some statements of the system would have to remain unprovable, or if provable, the system would be inconsistent. Many have argued that Gödel’s result showed that logicism must fail, or at least that some versions of it must fail, but it is important to add that the exact impact of his result on logicism is a complicated question, and subject to different interpretations. However this may be, all these discussions concern logic and mathematics as expressed through formalized symbolic systems, which were developed in the late nineteenth century, and in the twentieth century, and these discussions have had, in fact, no real effect on our ordinary reasoning in daily life, or on our everyday ability to add and subtract correctly.

Lest these last remarks seem philosophically controversial, let me say a bit more to explain them.

In Isaac Newton’s day, none of these formalized systems—systems of mathematical logic or of fully symbolic logic—existed, and yet hardly anyone would say, I think, that without these systems Newton was unable to add simple sums correctly or to draw logical inferences correctly. It follows that his ability to do these things was quite independent of such systems. More broadly, formal systems of logic and mathematics can certainly improve and refine our logical and mathematical abilities, but the abilities are already partially present in us without the systems, and it is precisely because we have some of these abilities antecedently that the formal systems can be constructed at all.

These same logical and mathematical abilities are also antecedent to our musings about ontology, or to our disagreements about ontology. Historically, there have been many different theories of what numbers are, just as there have been many different theories of what kinds of entities the propositions of logical argumentation are. Nevertheless, two and three have always made five, and the Barbara syllogism, at least in ordinary cases, has always been valid. Ontology is still a fascinating subject, to be sure, but its practical effects are often quite limited. Ontology certainly does affect higher mathematics, but there are large stretches of ordinary reasoning and arithmetical reckoning that are essentially immune to it.

So whatever results might be derived from various oddities in the further reaches of logic and mathematics, you can still be tolerably sure that, if all cats are cool, and Felix is a cat, then Felix is cool. And you can be equally sure that, if you have five rabbits, and you then add seven rabbits, you will have twelve rabbits, at least for a while.

Is there a role of mathematics in the development of human consciousness?

Is there a role of mathematics in the development of human consciousness?

In addition to Hofstadter's wonderful writings, you might also be interested in work done on the relationships between mathematics and cognition (more generally than just consciousness). Take a look at these classics in that area:

Rochel Gelman & C.R. Gallistel, The Child's Understanding of Number (Harvard University Press, 1986)

George Lakoff & Rafael Nuñez, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (Basic Books, 2001)

Stanislaus Dehaene, The Number Sense: How the Mind Creates Mathematics (Oxford University Press, 2011)

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 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.

Is mathematics independent of human consciousness?

Is mathematics independent of human consciousness?

I'm strongly inclined to say yes. Here's an argument. If there's even one technological civilization elsewhere in our unimaginably vast universe, then that civilization must have discovered enough math to produce technology. But we have no reason at all to think that it's a human civilization, given the very different conditions in which it evolved: if it exists, it belongs to a different species from ours. So: If math depends on human consciousness, then we're the only technological civilization in the universe, which seems very unlikely to me.

Here's a second argument. Before human beings came on the scene, did the earth orbit the sun in an ellipse, with the sun at one focus? Surely it did. (Indeed, there's every reason to think that the earth traced an elliptical orbit before any life at all emerged on it.) But "orbiting in an ellipse with the sun at one focus" is a precise mathematical description of the earth's behavior, a description that held true long before consciousness emerged here. Kepler may have discovered that description, but the truth of the description predated him and every other human. So at least one true mathematical description is independent of human consciousness.

Here's a third argument. If the answer to your question is no, then there were zero mathematical truths before human beings came along, in which case there weren't more than zero mathematical truths. But the fact that zero isn't more than zero is a mathematical truth. So there couldn't have been zero mathematical truths. So the answer to your question couldn't be no.

In writing mathematical proofs, I've been struck that direct proofs often seem

In writing mathematical proofs, I've been struck that direct proofs often seem to offer a kind of explanation for the theorem in question; an answer the question, "Why is this true?", as it were. By contrast, proofs by contradiction or indirect proofs often seem to lack this explanatory element, even if they they work just as well to prove the theorem. The thing is, I'm not sure it really makes sense to talk of mathematical "explanations." In science, explanations usually seem to involve finding some kind of mechanism behind a particular phenomenon or observation. But it isn't clear that anything similar happens in math. To take the opposing view, it seems plausible to suppose that all we can really talk about in math is logical entailment. And so, if both a direct and an indirect proof entail the theorem in question, it's a mistake to think that the former is giving us something that the latter is not. Do the panelists have any insight into this?

I probably should have noted before that, in the case of the different proofs of the first incompleteness theorem in Boolos, Burgess, and Jeffrey, the first proof they give is indirect or, as it is sometimes put, non-constructive: The proof shows us that, in any given consistent theory of sufficient strength, there is an "undecidable" sentence, one that is neither provable nor refutable by that theory; but the proof does not actually provide us with an example of an undecidable sentence.

The second proof, which is closer to Gödel's own, is direct and constructive: It does give us such a sentence, the so-called Gödel sentence for the theory. By doing so, it gives us more information than the first proof. It shows us, in particular, the there will always be an "undecidable sentence" of a very particular form (a so-called Π1 sentence).

This is a good example of why constructive proofs are often better than non-constructive proofs: They often give us more information. But it does not directly address the issue about explanation.

Hello philosophers,

Hello philosophers, I have yet another question. This time it's on the fundamental foundations of mathematics. I would like to know what Gödel's incompleteness theorem and inconsistency theorem actually stated. Intuitively, math seems logical, in the physical world, if you have two inanimate objects say two pencils laying on the table is it not logical that if you take one away you are only left with one on the table? An ex- professor of mine once told us in mathematics that ZF math was inconsistant and if we could prove that math does not work not only would we win a Fields Prize but we would also be the Herod of children all over the world ( assuming kids don't like to learn fundamental mathematics). Thank You again, Dale G.

You asked what Goedel's incompleteness and inconsistency theorems state.

Goedel proved two theorems known as his incompleteness theorems; I don't know of any called an "inconsistency" theorem (of course, he proved many other theorems, too!):

Informally, the first one--perhaps it is also the most famous one--says that any formal system that is based on first-order logic plus Peano's axioms for arithmetic is such that:

if it is consistent

(that is, if no contradiction can be proved in it),

then it is incomplete

(that is, there is some proposition P in the language of the system such that neither P nor not-P can be proved in the system;

presumably, only one of P and not-P is true; hence, there is some proposition in the language of the system that is true but unprovable in the system).

Even more informally, an English-language version of the true-but-unprovable sentence can be expressed thus:

This sentence is not provable.

(If it is false, then it is provable, hence true. So it can't be false. Hence it is true. Hence it is unprovable.)

Informally, the second theorem that Goedel proved says that if the system mentioned above is consistent, then its consistency cannot be proved in that system.

(You might be able to prove its consistency in some other system, however.)

You ask if it is logical that only one pencil would be left on a table if you had two pencils and removed one. Yes, it is. I'm not sure what that has to do with Goedel.

I have also never heard that ZF is inconsistent. I suppose that, if you could prove that it was, you might win the Fields medal. (But, even if you could prove that and did win the medal, I'm not sure that being "the Herod of children all over the world" would be a good thing, though being a hero of children might be :-)

For good introductions to Goedel's theorems, see Douglas Hofstadter's wonderful book Goedel Escher Bach, or Torkel Franzen's equally wonderful--but more technical--book Goedel's Theorem: An Incomplete Guide to Its Use and Abuse.