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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.

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,...

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.

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.

I agree with you that the sources of truth in mathematics can't be physical. For it seems clear to me that there would be mathematical truths even in a world that contained nothing physical at all (for instance, it would be true that the number of physical things in such a world is zero and therefore not greater than zero, not prime , etc.). So the sources of mathematical truth must be other than physical: if you like, metaphysical. Does this fact mean that all mathematicians are doing metaphysics? I don't think so. Metaphysicians can investigate the sources of truth in mathematics and the ontological status of mathematical truth-makers. But mathematicians themselves can simply make use of those truths without having to delve into what it is that makes those mathematical truths true.

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.

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....

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.

You've asked a terrific question! I wish I were more qualified to venture an answer to it. As you suggest, a sound direct proof of a theorem shows that the theorem must be true, in the broadest possible sense of "must." But a sound indirect proof shows the same thing. The difference, if any, seems purely psychological: some people find one proof psychologically more satisfying than the other. My sense is that some philosophers of math take this psychological difference very seriously and propose far-reaching revisions to classical math on the basis of it. You might take a look at the SEP entry on intutionism in the philosophy of math , particularly the discussion of constructive and nonconstructive proofs. The entry includes other helpful links and references too.

Imagine that a Greek philosopher promised to his queen that he would determine

Imagine that a Greek philosopher promised to his queen that he would determine the greatest prime number. He failed. Do you think that the mathematical fact that primes are infinite was a cause of his failure? I'm asking this because I guess most philosophers think that mathematical facts have no causal effects.

You've asked an interesting question, one related to what's often called the "Benacerraf problem" in the philosophy of mathematics (see section 3.4 of this SEP entry). I'm not sure that the problem is peculiar to mathematics. Imagine that the philosopher tried to impress his queen by creating a colorless red object. Was his failure caused by the fact that colorless red objects are impossible? If facts about color and facts about redness in particular can have causal power, can the fact that colorless red objects are impossible have causal power?

Part of the problem may be that these questions assume that we have a better philosophical grasp of the concept of fact and the concept of cause than we actually do. Given our currently poor grasp of those concepts, I don't think we should be confident that mathematical explanations or mathematical knowledge must depend on the causal power of mathematical facts.

You've asked an interesting question, one related to what's often called the "Benacerraf problem" in the philosophy of mathematics (see section 3.4 of this SEP entry ). I'm not sure that the problem is peculiar to mathematics. Imagine that the philosopher tried to impress his queen by creating a colorless red object. Was his failure caused by the fact that colorless red objects are impossible? If facts about color and facts about redness in particular can have causal power, can the fact that colorless red objects are impossible have causal power? Part of the problem may be that these questions assume that we have a better philosophical grasp of the concept of fact and the concept of cause than we actually do. Given our currently poor grasp of those concepts, I don't think we should be confident that mathematical explanations or mathematical knowledge must depend on the causal power of mathematical facts.

My understanding is that we can use systems like Peano Arithmetic to prove the

My understanding is that we can use systems like Peano Arithmetic to prove the seemingly basic truth that 1+1=2. Do such proofs actually give us reasons to believe that 1+1=2 that we didn't have before? Are they more fundamental or compelling than whatever justification a mathematically-naive person would have to believe that 1+1=2?

There are genuine philosophers of math on the Panel, but while we wait for them to respond I'll take a stab at your questions, which are epistemological as much as they're mathematical. I think we can answer yes to the first question without having to answer yes to the second question, but the answer to both questions may be yes.

As I understand the Peano Proof that 1 + 1 = 2, the gist is that the definitions of 'successor', 'addition', and '2' imply that 1 + 1 = 2. The successor of 1 is defined as 2, and addition is defined so that the result of adding 1 to any number is the successor of that number. Therefore, the result of adding 1 to 1 is 2. If the Peano Proof constitutes a reason to believe that 1 + 1 = 2, then it's surely a reason we didn't have before we had the Peano Proof. So I (somewhat tentatively) answer yes to your first question, regardless of the answer to your second question.

Even if we grant the infallibility of the deductive inferences in the Peano Proof, the Proof depends on Peano's postulates. So the answer to your second question depends on whether those postulates are 'more fundamental and compelling than whatever justification a mathematically naive person would have to believe that 1 + 1 = 2'. That, in turn, depends on what the mathematically naive person's justification is. In my answer to Question 5387, I argued that actual or possible processes of physical counting and aggregation are only fallible grounds for believing arithmetical claims such as 1 + 1 = 2. If those are the mathematically naive person's grounds, then they aren't as compelling as infallible grounds would be.

Are the Peano postulates infallible grounds? Some would regard that question as ill-posed because, they say, a postulate is just a stipulation and therefore not something that can be objectively true or false. But I'm not so sure. Euclid's Parallel Postulate helps to define Euclidean geometry, and in that sense one can regard it as merely a stipulation. But the Parallel Postulate is false if construed as a claim about any possible space and may actually be false as a claim about the physical space of our universe; in that sense it has an objective truth-value. So too might the Peano posulates have objective truth-values, in which case it makes sense to ask if they might be false. For what it's worth, I can't see how they could be false. But, as I said earlier, I can see how physical counting and aggregation might fail to confirm the claim that 1 + 1 = 2. So I tentatively answer yes to your second question as well.

There are genuine philosophers of math on the Panel, but while we wait for them to respond I'll take a stab at your questions, which are epistemological as much as they're mathematical. I think we can answer yes to the first question without having to answer yes to the second question, but the answer to both questions may be yes . As I understand the Peano Proof that 1 + 1 = 2 , the gist is that the definitions of 'successor', 'addition', and '2' imply that 1 + 1 = 2. The successor of 1 is defined as 2, and addition is defined so that the result of adding 1 to any number is the successor of that number. Therefore, the result of adding 1 to 1 is 2. If the Peano Proof constitutes a reason to believe that 1 + 1 = 2, then it's surely a reason we didn't have before we had the Peano Proof. So I (somewhat tentatively) answer yes to your first question, regardless of the answer to your second question. Even if we grant the infallibility of the deductive inferences in the Peano Proof, the...

I am interested in how mathematical propositions relate to objects in the world;

I am interested in how mathematical propositions relate to objects in the world; that is, how math and its concepts somehow correspond to the physical world. I have thought a bit about the issue, and realize that what happens, say, with numbers when we do some kind of mathematical operation with them may be the same as when we deduce one proposition in logic from another (If there is a number 2 and an operation "+", and an operation "=", then the result of using 2 + 2 = 4); but my question is this: does the proposition 2 + 2 = 4 mean the same thing as taking two objects and placing two more objects alongside of them, and then counting that there are four objects?

Philosophers continue to debate the relationship of mathematics to the physical world, including why mathematics is so effective at describing the physical world. The SEP entry on "Explanation in Mathematics," available at this link, contains much useful discussion as well as many references to further reading. At least one of the articles cited in the bibliography is available online: The Miracle of Applied Mathematics, by Mark Colyvan. I hope these prove helpful.

Strictly speaking, the proposition that 2 + 2 = 4 can't mean the same thing as the process of taking two objects, placing two more objects alongside them, and then counting that there are four objects in total. Propositions and processes belong to different categories. Moreover, one might doubt that the proposition that 2 + 2 = 4 even entails that whenever you take two physical objects and place two more physical objects alongside them, there will be four physical objects to count up. Why? Because the proposition that 2 + 2 = 4 is necessarily true: true in all possible worlds whatsoever. Yet there are strange-but-possible worlds where any two objects of a certain kind vanish whenever you try to put two more objects of that kind alongside them; even in those strange worlds, 2 + 2 = 4.

Indeed, even in the actual world 2 + 2 = 4 doesn't infallibly guide us about what happens when we put things together. Notoriously, if you mix 2 liters of water and 2 liters of ethanol, you get less than 4 liters of solution. Good to know if you're planning a party.

Philosophers continue to debate the relationship of mathematics to the physical world, including why mathematics is so effective at describing the physical world. The SEP entry on "Explanation in Mathematics," available at this link , contains much useful discussion as well as many references to further reading. At least one of the articles cited in the bibliography is available online: The Miracle of Applied Mathematics , by Mark Colyvan. I hope these prove helpful. Strictly speaking, the proposition that 2 + 2 = 4 can't mean the same thing as the process of taking two objects, placing two more objects alongside them, and then counting that there are four objects in total. Propositions and processes belong to different categories. Moreover, one might doubt that the proposition that 2 + 2 = 4 even entails that whenever you take two physical objects and place two more physical objects alongside them, there will be four physical objects to count up. Why?...

Does the fact that our perceptions can be represented geometrically and that

Does the fact that our perceptions can be represented geometrically and that geometry consists of eternal truths independent of the mind prove that an external reality underlies our perceptions?

I don't think that such an argument would rationally compel external-world skeptics (who say that no one can know that there's an external world) to abandon their view. External-world skeptics think that no one can know that solipsism is false, where solipsism is the claim that nothing external to oneself and one's mind exists. The solipsist won't grant that geometry consists of truths that are independent of his own mind, because he thinks nothing is. The solipsist could admit that his perceptions have a geometric character to them without having to attribute that character to something external. So I don't think solipsism can be disproven in the way you suggest.

All of this assumes that solipsism is otherwise intelligible. But one might argue that solipsism is unintelligible because it relies on the incoherent idea of a 'private language', an idea explored in detail in this SEP article.

I don't think that such an argument would rationally compel external-world skeptics (who say that no one can know that there's an external world) to abandon their view. External-world skeptics think that no one can know that solipsism is false, where solipsism is the claim that nothing external to oneself and one's mind exists. The solipsist won't grant that geometry consists of truths that are independent of his own mind, because he thinks nothing is. The solipsist could admit that his perceptions have a geometric character to them without having to attribute that character to something external. So I don't think solipsism can be disproven in the way you suggest. All of this assumes that solipsism is otherwise intelligible. But one might argue that solipsism is unintelligible because it relies on the incoherent idea of a 'private language', an idea explored in detail in this SEP article .

I have a question. Years ago me and two friends got into a debate about a riddle

I have a question. Years ago me and two friends got into a debate about a riddle. The riddle goes like this: A train starts from point A and is travelling towards point B. A wasp is travelling in the opposite direction at twice the speed of the train, the wasp touches the tip of the train and goes back to point B. How many times does the wasp touch the train? (this may be one version of many, but this is how it was told that faithful evening) So the "correct" answer was, infinte times. (similar to Zeno's paradox with Achilles and the tortoise). I said, well in theory it's infinte times, but if you were to actually do it, the train would hit point B eventually so it can't be infinte times? For it to be infinite times it would have to stop time (or something) So what would happen if you actually tried this? Say we do an experiment with a model train and instead of a wasp we use a laser (for accuracy). First we measure the railway track and only run the train, let's say it takes 10 seconds to go from...

I recommend reading the SEP entry on "Supertasks" available at this link. It contains helpful answers and references to further reading.

I recommend reading the SEP entry on "Supertasks" available at this link . It contains helpful answers and references to further reading.

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