Math is a language, not a science?

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Math is a language, not a science?

Postby sapmarten » Thu Jun 02, 2011 9:01 pm UTC

I think most here are familiar with Bertrand Russell's take on the issue. He contends that pure mathematics is the formalization of implications; for instance, math cannot prove the proposition x, nor can it prove the consequent implication y, but it can prove that x implies y assuming x to be true. Otherwise, math is an information communication device. It holds no claim towards demonstrating the truthfulness of the information within, but it allows extrapolation of the information and communication thereof. I think due to this that math is largely a language, albeit a very concise one. Science, however, is the collection and analysis of empirical data--by its very core, science is for demonstrating the truthfulness of information, not for communication or unsupported extrapolation.

I had always assumed that other mathematicians would agree on this issue, but I'd like to see how many actually do.

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Re: Math is a language, not a science?

Postby eSOANEM » Thu Jun 02, 2011 9:49 pm UTC

Maths is the science of consistent systems. Whether or not those systems are contained or have any relevance to the rest of the universe is irrelevant.

As I said in my post in the other thread (the one in science), science isn't one process, it's a collection of two or three concepts, that you must make predictions, that all predictions must be tested and that unpredicted phenomena should be explained.

In maths, all three of these apply. A conjecture is a prediction about the behaviour of whichever consistent system you're studying and it can be tested by finding a proof of its veracity (or lack thereof). The final concept, that of explaining unpredicted phenomena can be done by abstraction, generalisation or various other means (for example, if I discover that whenever I multiply a number by 7 and then divide by 7, I get the number I start with. The explanation for this "phenomenon" is that multiplication and division are each other's inverses).

That said, Maths can be used as a language as it is in science and engineering. The answer is that it's both, with pure maths being more like a science and less like a language and it's use in the sciences being less of a science itself and more of a language.
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Re: Math is a language, not a science?

Postby sapmarten » Thu Jun 02, 2011 9:54 pm UTC

...(for example, if I discover that whenever I multiply a number by 7 and then divide by 7, I get the number I start with. The explanation for this "phenomenon" is that multiplication and division are each other's inverses)...


If memory holds, I believe Mr. Russell contended that this was an axiom and not an observed "event". We defined this to be true as one of the axioms of logical implication--on the basis that mathematics was invented, not discovered.

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Re: Math is a language, not a science?

Postby Nat » Thu Jun 02, 2011 10:39 pm UTC

Yeah, Russel sounds about right. though math can say "X is true", for example if X is of the form "Z implies Y", which might suggest that math is also a science, so you need to define the difference more clearly.

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Re: Math is a language, not a science?

Postby skeptical scientist » Thu Jun 02, 2011 11:39 pm UTC

sapmarten wrote:[Bertrand Russell] contends ... math cannot prove the proposition x, nor can it prove the consequent implication y, but it can prove that x implies y.

This is nonsense, as Nat already pointed out. It's ridiculous to say that math cannot prove propositions, and then go on to say that math can prove propositions, if they take the form x implies y. I would assert it can also prove propositions that are not implications; for instance, you can prove existential statements like, "There exists an algebraically closed field extending the rationals."

Math is a language, but it is not just a language. It can do more than express things, it can also be used to make discoveries. It has this in common with a science. However, I would say it is not exactly a science, because it doesn't use the scientific method, and I think something has to use the scientific method to really be considered a science. Instead, math has its own standard for what constitutes proof.
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Re: Math is a language, not a science?

Postby nachomancer » Fri Jun 03, 2011 12:45 am UTC

eSOANEM wrote:Maths is the science of consistent systems.


This might be considered a minor quibble (and isn't relevant to the OP's question), but this seems too restrictive to me. If I found that a given axiomatic system was inconsistent, but that, say, the shortest contradiction was of a particular length, this would certainly constitute a valid/useful/interesting mathematical statement. In other words, it doesn't seem impossible that we could still ask interesting questions about inconsistent systems. Maybe a better definition would be that math is the science of formalizedd systems.

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Re: Math is a language, not a science?

Postby Tirian » Fri Jun 03, 2011 1:46 am UTC

sapmarten wrote:
...(for example, if I discover that whenever I multiply a number by 7 and then divide by 7, I get the number I start with. The explanation for this "phenomenon" is that multiplication and division are each other's inverses)...


If memory holds, I believe Mr. Russell contended that this was an axiom and not an observed "event". We defined this to be true as one of the axioms of logical implication--on the basis that mathematics was invented, not discovered.


Yeah, not so much an axiom as the definition of division is what makes this example a particularly short and intuitive proof.

An example that might divide Russell and other formalists from the larger community more firmly is the proposition that multiplication in the natural numbers is commutative. Russell certainly wouldn't have been satisfied with the notion that we've been using it for this long and never came up with a counterexample. (Nor should he -- he is rightfully well-known for dealing a mortal wound to naive set theory by thinking of a set no-one had thought of before that didn't behave in the way of the others.) An obvious second attempt would be to note that multiplication is modeled in the real world by the area of a rectangle whose height and width were the multiplier and the multiplicand and that the commutative rule follows as long as one is flexible of which of the alternate sides of a rectangle is the height and which is the width, but a formalist would similarly see that as highly inelegant. Instead, what Russell would want to do (indeed, what he DID do in the Principia Mathematica) is to formally define numbers and then the principles of induction and recursion and then define addition and multiplication using recursion and then inductively prove that multiplication is commutative based on those axioms. As I recall, it is a remarkably delicate proof that leaves one more willing to trust the intuitionists next time.

As to the basic question of whether math is a science, it will depend entirely on your definition of the words. As the OP defines sciences as the study of empirical data, then it obviously is not from that perspective. Alternatively, if one defines a science as an intellectual pursuit that is furthered by greater rationality (in contrast to "the arts" which are furthered by greater aesthetics), then math is certainly mostly (but not entirely) scientific in nature.

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Re: Math is a language, not a science?

Postby eSOANEM » Fri Jun 03, 2011 6:54 am UTC

nachomancer wrote:
eSOANEM wrote:Maths is the science of consistent systems.


This might be considered a minor quibble (and isn't relevant to the OP's question), but this seems too restrictive to me. If I found that a given axiomatic system was inconsistent, but that, say, the shortest contradiction was of a particular length, this would certainly constitute a valid/useful/interesting mathematical statement. In other words, it doesn't seem impossible that we could still ask interesting questions about inconsistent systems. Maybe a better definition would be that math is the science of formalizedd systems.


This is relevant. If the system is no longer consistent, your axioms must contain a contradiction and so any derived statement is ultimately meaningless as the opposite could also be "proven". It may be interesting to note that two axioms are contradictory, but other than that, nothing within that system has any meaning.
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Re: Math is a language, not a science?

Postby jestingrabbit » Fri Jun 03, 2011 8:20 am UTC

eSOANEM wrote:
nachomancer wrote:
eSOANEM wrote:Maths is the science of consistent systems.


This might be considered a minor quibble (and isn't relevant to the OP's question), but this seems too restrictive to me. If I found that a given axiomatic system was inconsistent, but that, say, the shortest contradiction was of a particular length, this would certainly constitute a valid/useful/interesting mathematical statement. In other words, it doesn't seem impossible that we could still ask interesting questions about inconsistent systems. Maybe a better definition would be that math is the science of formalizedd systems.


This is relevant. If the system is no longer consistent, your axioms must contain a contradiction and so any derived statement is ultimately meaningless as the opposite could also be "proven". It may be interesting to note that two axioms are contradictory, but other than that, nothing within that system has any meaning.


Yes, inconsistent sets of axioms entail explosion. However, according to you, Godel proved that if the Peano axioms are consistent, then we can't know if we're doing mathematics when we study the system described by the peano axioms. That doesn't make sense to me.

I don't think its useful to restrict mathematics to only the study of consistent systems when we can't know whether reasonably complicated systems are consistent. Furthermore, think that demonstrating a contradiction in a logical system is a mathematical act. Are you comfortable contending that when Russell demonstrated that naive set theory was inconsistent he wasn't doing mathematics?
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Re: Math is a language, not a science?

Postby eSOANEM » Fri Jun 03, 2011 5:25 pm UTC

jestingrabbit wrote:
eSOANEM wrote:
nachomancer wrote:
eSOANEM wrote:Maths is the science of consistent systems.


This might be considered a minor quibble (and isn't relevant to the OP's question), but this seems too restrictive to me. If I found that a given axiomatic system was inconsistent, but that, say, the shortest contradiction was of a particular length, this would certainly constitute a valid/useful/interesting mathematical statement. In other words, it doesn't seem impossible that we could still ask interesting questions about inconsistent systems. Maybe a better definition would be that math is the science of formalizedd systems.


This is relevant. If the system is no longer consistent, your axioms must contain a contradiction and so any derived statement is ultimately meaningless as the opposite could also be "proven". It may be interesting to note that two axioms are contradictory, but other than that, nothing within that system has any meaning.


Yes, inconsistent sets of axioms entail explosion. However, according to you, Godel proved that if the Peano axioms are consistent, then we can't know if we're doing mathematics when we study the system described by the peano axioms. That doesn't make sense to me.

I don't think its useful to restrict mathematics to only the study of consistent systems when we can't know whether reasonably complicated systems are consistent. Furthermore, think that demonstrating a contradiction in a logical system is a mathematical act. Are you comfortable contending that when Russell demonstrated that naive set theory was inconsistent he wasn't doing mathematics?


1. I'm not sure I understand what you're saying.

2. In the same way that disproving a scientific theory is an act of science yes. However, like proving a scientific theory false (as opposed to being accurate only in a limited domain), that theory doesn't fulfil the purpose of science (to describe the world) and, mathematically, it no longer describes a consistent system.
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Re: Math is a language, not a science?

Postby nachomancer » Fri Jun 03, 2011 11:21 pm UTC

eSOANEM wrote:This is relevant. If the system is no longer consistent, your axioms must contain a contradiction and so any derived statement is ultimately meaningless as the opposite could also be "proven". It may be interesting to note that two axioms are contradictory, but other than that, nothing within that system has any meaning.


This is obviously true, but you've missed my point. Imagine, for instance, that I have shown that a certain class of axiomatic systems (perhaps including, say peano arithmatic) contains only inconsistent systems, but that the shortest contradiction has length (according to the metric given by number of applications of an axiom or something) far larger than the size of the universe. We could still make meaningful statements as long as they are shorter than the shortest contradiction! Even if Peano arithmetic is proved inconsistent, I will still feel safe balancing my checkbook (using theorems of Peano arithmetic), for instance. Whether this particular instance occurs or not, it seems premature to rule out the possibility of, say, a classification of all axiomatic systems according to some measure of inconsistency degree or something. Therefore, the study of formally inconsistent systems is just as much mathematics as anything else.

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Re: Math is a language, not a science?

Postby Sheikh al-Majaneen » Sat Jun 04, 2011 12:31 am UTC

sapmarten wrote:
...(for example, if I discover that whenever I multiply a number by 7 and then divide by 7, I get the number I start with. The explanation for this "phenomenon" is that multiplication and division are each other's inverses)...


If memory holds, I believe Mr. Russell contended that this was an axiom and not an observed "event". We defined this to be true as one of the axioms of logical implication--on the basis that mathematics was invented, not discovered.

Actually, he and Alfred North Whitehead set out to prove that 1+1=2 over the course of about 500 pages in Principia Mathematica. Taking that as far as proving 7*7/7 would probably have taken several hundred or thousand more.

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Re: Math is a language, not a science?

Postby Tirian » Sat Jun 04, 2011 12:57 am UTC

Sheikh al-Majaneen wrote:
sapmarten wrote:
...(for example, if I discover that whenever I multiply a number by 7 and then divide by 7, I get the number I start with. The explanation for this "phenomenon" is that multiplication and division are each other's inverses)...


If memory holds, I believe Mr. Russell contended that this was an axiom and not an observed "event". We defined this to be true as one of the axioms of logical implication--on the basis that mathematics was invented, not discovered.

Actually, he and Alfred North Whitehead set out to prove that 1+1=2 over the course of about 500 pages in Principia Mathematica. Taking that as far as proving 7*7/7 would probably have taken several hundred or thousand more.


Not really. First, it was only on page 375 that they showed that the union of two disjoint singleton sets was a doubleton. Second, they started from type theory and didn't hand-wave a single proof. Third, I'm not going to check but I highly doubt the proof required anywhere near all 375 of those pages, if you know what I mean.

Finally, the proof that 7x/7 = x for all natural x is trivial once you get far enough in the book that you've actually defined multiplication and division. We define a/b to be the value c that satisfies ac=b if that equation has exactly one solution. So all you need to do is to show that left-multiplication is injective (which is an easy consequence of the easy fact that left-multiplication by non-zero numbers is a strict order-morphism on the naturals). Therefore, 7z=7x has the unique solution z=x, thus 7x/7=x.

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Re: Math is a language, not a science?

Postby eSOANEM » Sat Jun 04, 2011 6:38 am UTC

nachomancer wrote:
eSOANEM wrote:This is relevant. If the system is no longer consistent, your axioms must contain a contradiction and so any derived statement is ultimately meaningless as the opposite could also be "proven". It may be interesting to note that two axioms are contradictory, but other than that, nothing within that system has any meaning.


This is obviously true, but you've missed my point. Imagine, for instance, that I have shown that a certain class of axiomatic systems (perhaps including, say peano arithmatic) contains only inconsistent systems, but that the shortest contradiction has length (according to the metric given by number of applications of an axiom or something) far larger than the size of the universe. We could still make meaningful statements as long as they are shorter than the shortest contradiction! Even if Peano arithmetic is proved inconsistent, I will still feel safe balancing my checkbook (using theorems of Peano arithmetic), for instance. Whether this particular instance occurs or not, it seems premature to rule out the possibility of, say, a classification of all axiomatic systems according to some measure of inconsistency degree or something. Therefore, the study of formally inconsistent systems is just as much mathematics as anything else.


Hmmm... That's an interesting proposal. The first thing is that it would be difficult if not impossible to compare the size of the universe with the length of the shortest contradiction, but that's also not really the point.

My main concern is that if a set of axioms are contradictory, that there is no shortest length of contradiction (although this is merely hunch, I can't prove it) and any contradiction can just be reduced to two (or maybe more) of the axioms being contradictory without needing repeated application (again, I can't prove this, it just seems more intuitive to me).

If my intuition is wrong however, then the study of certain broadly-consistent inconsistent (if that makes sense. I mean system with long times to contradiction) systems would be maths and I'd have to revise my definition.
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Re: Math is a language, not a science?

Postby Kurushimi » Sat Jun 04, 2011 8:08 am UTC

nachomancer wrote:
eSOANEM wrote:This is relevant. If the system is no longer consistent, your axioms must contain a contradiction and so any derived statement is ultimately meaningless as the opposite could also be "proven". It may be interesting to note that two axioms are contradictory, but other than that, nothing within that system has any meaning.


This is obviously true, but you've missed my point. Imagine, for instance, that I have shown that a certain class of axiomatic systems (perhaps including, say peano arithmatic) contains only inconsistent systems, but that the shortest contradiction has length (according to the metric given by number of applications of an axiom or something) far larger than the size of the universe. We could still make meaningful statements as long as they are shorter than the shortest contradiction! Even if Peano arithmetic is proved inconsistent, I will still feel safe balancing my checkbook (using theorems of Peano arithmetic), for instance. Whether this particular instance occurs or not, it seems premature to rule out the possibility of, say, a classification of all axiomatic systems according to some measure of inconsistency degree or something. Therefore, the study of formally inconsistent systems is just as much mathematics as anything else.


That still doesn't make sense. Your axioms being inconsistent will make them useless. Whether or not I can reach a contradiction in less than n steps doesn't matter. The contradiction exists, and that's all that matters. What you're describing seems to be like saying "It all makes sense as long as you don't think about it too hard."

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Re: Math is a language, not a science?

Postby nachomancer » Sat Jun 04, 2011 8:51 am UTC

Kurushimi wrote:That still doesn't make sense. Your axioms being inconsistent will make them useless. Whether or not I can reach a contradiction in less than n steps doesn't matter. The contradiction exists, and that's all that matters. What you're describing seems to be like saying "It all makes sense as long as you don't think about it too hard."


Most of us use naive set thoery all the time in math, even though we know it is inconsistent. This doesn't mean that anything we prove with naive set theory is meaningless. As long as we avoid the contradictions, we get valid results (ie. we can't prove both a statement and its negation). This a little different than the symbolic length thing (which was pure conjecture on my part, I have no idea if its a useful notion or not), but the idea is the same: if you avoid the contradictions, you can get useful results even from inconsistent systems.

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Re: Math is a language, not a science?

Postby Kurushimi » Sat Jun 04, 2011 9:11 am UTC

nachomancer wrote:
Kurushimi wrote:That still doesn't make sense. Your axioms being inconsistent will make them useless. Whether or not I can reach a contradiction in less than n steps doesn't matter. The contradiction exists, and that's all that matters. What you're describing seems to be like saying "It all makes sense as long as you don't think about it too hard."


Most of us use naive set thoery all the time in math, even though we know it is inconsistent. This doesn't mean that anything we prove with naive set theory is meaningless. As long as we avoid the contradictions, we get valid results (ie. we can't prove both a statement and its negation). This a little different than the symbolic length thing (which was pure conjecture on my part, I have no idea if its a useful notion or not), but the idea is the same: if you avoid the contradictions, you can get useful results even from inconsistent systems.


Things are explained using naive set theory because it's intuitive. These things can also be proven with a consistent set theory. But if we can't prove it with a consistent theory it is meaningless. Saying "if you avoid contradiction you can still get useful results" is ludicrous because if you can reach contradictions at all there's something wrong with your system. Proofs in an inconsistent system are meaningless because I can give counterproofs in the same system.

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Re: Math is a language, not a science?

Postby skeptical scientist » Sat Jun 04, 2011 12:06 pm UTC

nachomancer wrote:Most of us use naive set thoery all the time in math, even though we know it is inconsistent. This doesn't mean that anything we prove with naive set theory is meaningless. As long as we avoid the contradictions, we get valid results (ie. we can't prove both a statement and its negation). This a little different than the symbolic length thing (which was pure conjecture on my part, I have no idea if its a useful notion or not), but the idea is the same: if you avoid the contradictions, you can get useful results even from inconsistent systems.

Well, the only difference between naive set theory and ZFC is that the axiom of unrestricted comprehension is replaced by restricted comprehension, so I don't think this is true. People think in terms of naive set theory, but (those who are aware of the paradoxes) restrict their uses of comprehension, so they're really working (informally) within ZFC.
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Re: Math is a language, not a science?

Postby Token » Sat Jun 04, 2011 12:40 pm UTC

skeptical scientist wrote:Well, the only difference between naive set theory and ZFC is that the axiom of unrestricted comprehension is replaced by restricted comprehension, so I don't think this is true. People think in terms of naive set theory, but (those who are aware of the paradoxes) restrict their uses of comprehension, so they're really working (informally) within ZFC.

I think you may be mis-stating the gap between formal systems and everyday mathematics. Naive set theory doesn't *have* axioms, let alone an axiom of unrestricted comprehension. The reason axiomatic systems were developed in the first place was to deal with problems arising from this (Russell's paradox being the canonical example). I think what you're meaning to say is that most mathematicians are aware of and avoid the basic problems (such as talking about "the set of all sets that satisfy property X"), and that current usage of naive set theory is therefore *translatable* into ZFC. The difference is small, but significant - most people like to do mathematics without, say, stating an instance of the axiom schema of comprehension (restricted or unrestricted), then proceeding via tautologies and modus ponens.
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Re: Math is a language, not a science?

Postby skeptical scientist » Sat Jun 04, 2011 1:04 pm UTC

Token wrote:The difference is small, but significant - most people like to do mathematics without, say, stating an instance of the axiom schema of comprehension (restricted or unrestricted), then proceeding via tautologies and modus ponens.

That's what I meant when I said they were working informally - i.e. not in terms of formal deductions. But it really is basically the same thing. When we say "let S = {f : f is continuous and has compact support}" we really are using the comprehension axiom; when we prove A subset B and B subset A and conclude A=B we're using extensionality; and so forth, even if we're not consciously thinking in those terms.
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Re: Math is a language, not a science?

Postby Token » Sat Jun 04, 2011 2:52 pm UTC

skeptical scientist wrote:When we say "let S = {f : f is continuous and has compact support}" we really are using the comprehension axiom; when we prove A subset B and B subset A and conclude A=B we're using extensionality; and so forth, even if we're not consciously thinking in those terms.

Not at all. If I move from A is a subset of B and B is a subset of A to A equals B, then I'm using an abstract semantic notion of what it means to be a set (I'm viewing a set as an abstract collection of objects, and using my intuition that such collections are essentially unique up to the objects that they contain). Using the axiom of extensionality means that I've taken [imath]A \subseteq B[/imath] and [imath]B \subseteq A[/imath] and proved [imath]A = B[/imath] to be a syntactic consequence of those two statements, the axiom of extensionality, and any relevant definitions (using whatever underlying logic I may have chosen). The fact that the two tie in to each other is related to the fact that the axiom of extensionality, as a string of symbols, has a semantic interpretation that matches my intuition about sets - this is why it's used in the first place! But this semantic interpretation is entirely independent of the particular syntactic representation I've chosen - there are other syntactically different axioms (some even semantically different, up to a point) that have the same semantic consequences, and I could equally well have used any of them.

My point being that, in general, doing mathematics involves using a (somewhat formalized) intuition about what mathematics actually is. The fact that we have a formal system that we believe represents some or all of that intuition does *not* mean that doing mathematics is intrinsically linked to that particular formal system. I mean, mathematicians existed before the twentieth century who managed to prove plenty of valid results, using the same general principles we use today, and you'd have to stretch to claim that *they* were using ZFC.
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Re: Math is a language, not a science?

Postby skullturf » Sat Jun 04, 2011 4:40 pm UTC

My $0.02: I find myself siding with Token and Nachomancer. I don't think inconsistent systems are necessarily useless, and I have some doubts as to whether "doing mathematics" is synonymous with "doing stuff that can be easily translated into a formal system".

All the mathematics that people like Euler and Fourier did in the 1700s and 1800s was still mathematics, even though it wasn't formalized and definitions weren't always precise. (However, I would certainly say the later rigorization of analysis was a huge intellectual advance, and an improvement of mathematics.)

Perhaps more importantly, I don't think we actually know that contemporary mathematics (as formalized in ZFC, say) is consistent. I think it's very very very unlikely that ZFC is inconsistent, but we can't prove that it's consistent. In principle, for all we know, right now we could be in the same situation that Frege was in right before he received his letter from Bertrand Russell.

I haven't read Frege's work, so this is pure speculation. But I bet you that even though it rests on inconsistent premises, there are probably particular arguments in there that can be "salvaged" to work in another system.

If somebody finds a contradiction in ZFC next year, I think what would happen is that with time, we'd come up with other axioms for set theory, and we'd find that the vast majority of what we call "mathematics" could be proved in the new system as well.

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Re: Math is a language, not a science?

Postby skeptical scientist » Sat Jun 04, 2011 4:57 pm UTC

Fine, if you want to think of ZFC as a purely syntactic thing, go ahead. I think of it as both a semantic and a syntactic object, which happen to coincide, so that e.g. the axiom of extensionality says, "Sets are determined by membership," rather than, "∀A∀B (∀C(C ϵ A <->C ϵ B) -> A = B)." But I find such debates tiring and ultimately futile.

So rather than have this debate, which completely misses the point of what I was saying, please go back and amend my earlier post so that instead of saying "the only difference between naive set theory and ZFC is restricted vs. unrestricted choice", pretend I said
skepsci' wrote:The only thing you need to do to make naive set theory consistent is be careful about what properties determine sets, basically using a semantic version of restricted comprehension rather than unrestricted comprehension. This is what people do in practice anyways, so it's not really true to say mathematicians prove things using an inconsistent version of naive set theory.
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Re: Math is a language, not a science?

Postby Kurushimi » Sat Jun 04, 2011 6:08 pm UTC

skullturf wrote:My $0.02: I find myself siding with Token and Nachomancer. I don't think inconsistent systems are necessarily useless, and I have some doubts as to whether "doing mathematics" is synonymous with "doing stuff that can be easily translated into a formal system".

All the mathematics that people like Euler and Fourier did in the 1700s and 1800s was still mathematics, even though it wasn't formalized and definitions weren't always precise. (However, I would certainly say the later rigorization of analysis was a huge intellectual advance, and an improvement of mathematics.)


You don't seem to appreciate the importance of logical rigour. Sure, we can be hand-wavy about certain facts and still arrive at true statements. But it's also true that we can be hand-wavy about certain facts and arrive at ridiculous statements as a result. Most of the time a lack of logical rigour won't cause a problem but there are times when it does. Just because something works most of the time doesn't mean there isn't a problem with it.

Secondly, if a logical system is inconsistent, we can prove both "A is true" and "A is not true." Since both proofs were arrived at in the same way, why should one have any more merit over the other?

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Re: Math is a language, not a science?

Postby Token » Sat Jun 04, 2011 6:45 pm UTC

skeptical scientist wrote:Fine, if you want to think of ZFC as a purely syntactic thing, go ahead. I think of it as both a semantic and a syntactic object, which happen to coincide, so that e.g. the axiom of extensionality says, "Sets are determined by membership," rather than, "∀A∀B (∀C(C ϵ A <->C ϵ B) -> A = B)." But I find such debates tiring and ultimately futile.

So rather than have this debate, which completely misses the point of what I was saying, please go back and amend my earlier post so that instead of saying "the only difference between naive set theory and ZFC is restricted vs. unrestricted choice", pretend I said
skepsci' wrote:The only thing you need to do to make naive set theory consistent is be careful about what properties determine sets, basically using a semantic version of restricted comprehension rather than unrestricted comprehension. This is what people do in practice anyways, so it's not really true to say mathematicians prove things using an inconsistent version of naive set theory.

I don't think of ZFC as a purely syntactic object. I mean, it is, but taking it on its own would be kind of missing the point. What makes it interesting is the interpretation that we give - that it can be thought of as representing set theory. And while ZFC is useless without such an interpretation (and probably useless without this particular interpretation), set theory itself does not require a syntactic representation to be useful. So yes, it's pretty much pointless to consider the syntax of ZFC without the set-theoretic semantics, but it's possible to talk about set theory as a concept without tying it down to a single (or any) formal system.

So saying "mathematicians are using ZFC" when you actually mean "mathematicians are using set theory" is a mistake that's worth correcting, especially in the context of this kind of discussion. In general, mathematicians don't deliberately reason about only those sets whose existence is provable in ZFC - they reason about whatever sets are currently relevant to them. It happens that in general either the two coincide, or it's irrelevant whether a particular class of objects is a set or not, which is why ZFC is commonly used instead of being rejected as a bad formalization of set theory. Your rewording still makes this error - changing "axiom" to "semantic version" doesn't fix your assertion, it just makes it meaningless. What exactly is a "semantic version of restricted comprehension", anyway?

Kurushimi wrote:You don't seem to appreciate the importance of logical rigour. Sure, we can be hand-wavy about certain facts and still arrive at true statements. But it's also true that we can be hand-wavy about certain facts and arrive at ridiculous statements as a result. Most of the time a lack of logical rigour won't cause a problem but there are times when it does. Just because something works most of the time doesn't mean there isn't a problem with it.

It's possible to work in a formal system and still arrive at ridiculous statements - all it takes is for the system to be inconsistent. And we can't know whether a formal system is consistent (at least, not if its interesting), so being more rigorous doesn't solve the problem.
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Re: Math is a language, not a science?

Postby skullturf » Sat Jun 04, 2011 7:11 pm UTC

Kurushimi wrote:But it's also true that we can be hand-wavy about certain facts and arrive at ridiculous statements as a result. Most of the time a lack of logical rigour won't cause a problem but there are times when it does. Just because something works most of the time doesn't mean there isn't a problem with it.

I agree with that part.

I'm just saying that imprecise or nonrigorous mathematics is not literally 100% useless.

Inconsistent systems can have some value. We might be using one now! (I don't think that's at all likely, but it's possible.)

Kurushimi wrote:Secondly, if a logical system is inconsistent, we can prove both "A is true" and "A is not true." Since both proofs were arrived at in the same way, why should one have any more merit over the other?

Well, once we discover proofs of both "A" and "not A" in the same system, then of course we should no longer think of the system as a reliable way of generating true statements.

But I think what would probably happen in practice is that many of the things we proved in that system would be "salvageable".

We could come up with a new formal system whose consistency we have more confidence in (even if we can't prove it), and we would probably find that many of our proofs could be adapted to the new system.

As opposed to throwing absolutely everything away and literally starting all of mathematics over again from scratch.

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Re: Math is a language, not a science?

Postby skeptical scientist » Sat Jun 04, 2011 7:21 pm UTC

Token wrote:So saying "mathematicians are using ZFC" when you actually mean "mathematicians are using set theory" is a mistake that's worth correcting, especially in the context of this kind of discussion. In general, mathematicians don't deliberately reason about only those sets whose existence is provable in ZFC - they reason about whatever sets are currently relevant to them. It happens that in general either the two coincide, or it's irrelevant whether a particular class of objects is a set or not, which is why ZFC is commonly used instead of being rejected as a bad formalization of set theory. Your rewording still makes this error - changing "axiom" to "semantic version" doesn't fix your assertion, it just makes it meaningless. What exactly is a "semantic version of restricted comprehension", anyway?

I'm saying that we can't and don't accept that all collections are sets. There is no set of all sets, for example, nor a set of all ordinals, or groups. Certain collections are sets, and other collections are proper classes. When I say mathematicians accept a semantic version of restricted comprehension, I'm saying that {x ϵ A : P(x)} is always a set, but {x : P(x)} is not necessarily a set for arbitrary P.

Further, I didn't use "mathematicians are using ZFC" when I meant "mathematicians are using set theory". I meant that mathematicians are using a particular version of set theory where arbitrary collections are not necessarily considered sets (in particular, some collections are proper classes), and the operations that produce collections which are sets more or less coincide with the axioms of ZFC—e.g. restricted comprehension, replacement, and power set. And while mathematicians may not reason solely about sets whose existence is provable in ZFC, they won't reason about "sets" whose non-existence is provable in ZFC. They may reason about proper classes, but they won't try to answer questions which only apply to sets. (e.g. "What is the cardinality of this class?")
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Re: Math is a language, not a science?

Postby Token » Sat Jun 04, 2011 7:47 pm UTC

Ah, I see. What you actually mean is that they reject unrestricted comprehension. I think I mostly agree with you, then.
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Re: Math is a language, not a science?

Postby skeptical scientist » Sat Jun 04, 2011 8:26 pm UTC

Token wrote:Ah, I see. What you actually mean is that they reject unrestricted comprehension. I think I mostly agree with you, then.

Yes, but more than that. I suspect it's no accident that common instances of set-builder notation {x ϵ A : P(x)} and {f(x) : x ϵ A} are exactly mirrored by the axioms of restricted comprehension and replacement. Without unrestricted comprehension, you need some way to say "ok, this set is actually a set," and that's where ZFC comes in: to fill the void left by removing unrestricted comprehension.
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Re: Math is a language, not a science?

Postby nachomancer » Sat Jun 04, 2011 9:26 pm UTC

Kurushimi wrote:Things are explained using naive set theory because it's intuitive. These things can also be proven with a consistent set theory. But if we can't prove it with a consistent theory it is meaningless. Saying "if you avoid contradiction you can still get useful results" is ludicrous because if you can reach contradictions at all there's something wrong with your system. Proofs in an inconsistent system are meaningless because I can give counterproofs in the same system.


Consider the system ZFC+Axiom X, where axiom X is something like "For all P, P and not P" (assume for the sake of argument that ZFC is consistent). Any standard proof in ZFC also works in this new system, and they will stay consistent as long as we don't use Axiom X. Obviously this system is inconsistent, but it is not true that all of its theorems are "meaningless". This is because some of them (the ones not using X) still hold when we modify our axioms to exclude axiom X (our contradiction). You seem to be saying that any result proved in a system that was later shown to be inconsistent (ie all of math before the 20th century) is meaningless. This is not true, because as long as the proofs avoided certain combinations of axioms (ie assuming the existence of a set of all sets, etc.) they still hold once we've modified our axioms to exclude the paradoxes.

The point of my example (ZFC+X) is that working in ZFC and working in ZFC+X but avoiding using X are the same thing, just like working in ZFC and working in naive set theory, but avoiding contradictions, are the same. It seems intuitively plausible that there should be some sense in which naive set theory is "more consistent" than the system consisting of just axiom X (for instance). Given any inconsistent system, if I can distinguish some contradiction free area (ie all statements of length less than n for instance) than I can clearly modify my system to a different one that includes only this "good" area. My point is that we can make the distinction between "globally consistent" (no contradictions anywhere) and "locally consistent" (no contradictions nearby) systems, even if it is not usually done.

(If you don't think naive set theory can be axiomatized, just substitute any axiomatization of set theory that was later shown to be inconsistent)

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Re: Math is a language, not a science?

Postby Kurushimi » Sun Jun 05, 2011 8:31 am UTC

nachomancer wrote:
Kurushimi wrote:Things are explained using naive set theory because it's intuitive. These things can also be proven with a consistent set theory. But if we can't prove it with a consistent theory it is meaningless. Saying "if you avoid contradiction you can still get useful results" is ludicrous because if you can reach contradictions at all there's something wrong with your system. Proofs in an inconsistent system are meaningless because I can give counterproofs in the same system.


Consider the system ZFC+Axiom X, where axiom X is something like "For all P, P and not P" (assume for the sake of argument that ZFC is consistent). Any standard proof in ZFC also works in this new system, and they will stay consistent as long as we don't use Axiom X. Obviously this system is inconsistent, but it is not true that all of its theorems are "meaningless". This is because some of them (the ones not using X) still hold when we modify our axioms to exclude axiom X (our contradiction). You seem to be saying that any result proved in a system that was later shown to be inconsistent (ie all of math before the 20th century) is meaningless. This is not true, because as long as the proofs avoided certain combinations of axioms (ie assuming the existence of a set of all sets, etc.) they still hold once we've modified our axioms to exclude the paradoxes.


That doesn't in anyway contradict what I said. Proofs in ZFC + Axiom X are meaningless. I can prove the moon is made of cheese in ZFC + Axiom X. Proofs in ZFC are not meaningless.

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Re: Math is a language, not a science?

Postby genevabeagle » Sun Jun 05, 2011 11:27 am UTC

Restricting "mathematics" to take place only within consistent systems seems a little bit excessive. Part of dealing with consistent systems is differentiating them from inconsistent systems, and even finding ways to disentangle the two if they collide somehow. This doesn't change the fact that once a system is known to be inconsistent, continuing to prove things using that system is obviously not doing meaningful math.

But poking around inside an inconsistent system, to see what could be twerked to make the thing into a (probably) consistent system, strikes me as a mathematical activity. I don't think it typically involves a formalized system of reasoning, but I wouldn't really categorize serious axiom-level tinkering as anything other than mathematical.

I see I've been beaten to the punch:
nachomancer wrote:In other words, it doesn't seem impossible that we could still ask interesting questions about inconsistent systems. Maybe a better definition would be that math is the science of formalized systems.

jestingrabbit wrote:I don't think it's useful to restrict mathematics to only the study of consistent systems when we can't know whether reasonably complicated systems are consistent. Furthermore, I think that demonstrating a contradiction in a logical system is a mathematical act.

Edit: corrected a misattribution.
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Re: Math is a language, not a science?

Postby skullturf » Sun Jun 05, 2011 5:22 pm UTC

Kurushimi wrote:
nachomancer wrote:Consider the system ZFC+Axiom X, where axiom X is something like "For all P, P and not P" (assume for the sake of argument that ZFC is consistent). Any standard proof in ZFC also works in this new system, and they will stay consistent as long as we don't use Axiom X. Obviously this system is inconsistent, but it is not true that all of its theorems are "meaningless".

That doesn't in anyway contradict what I said. Proofs in ZFC + Axiom X are meaningless. I can prove the moon is made of cheese in ZFC + Axiom X. Proofs in ZFC are not meaningless.

That deserves to be stated more carefully. Is "meaningless" really the best word for describing proofs in an inconsistent system?

I completely agree that as soon as you find one contradiction in a formal system, you know the system cannot be relied on as a method of generating true statements. I probably would have been more OK with the word "useless" than "meaningless" (though it could be argued that experimenting with proofs in inconsistent systems can be useful too).

Proofs in an inconsistent system can be more than "meaningless", in the sense that individual statements in the proof can still have meaning.

Anyway, I realize we may be mostly just quibbling over words here, and I hesitate to be too pedantic, but then again, this is a web forum where we're talking about the philosophy of mathematics.

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Re: Math is a language, not a science?

Postby eSOANEM » Sun Jun 05, 2011 8:36 pm UTC

skullturf wrote:
Kurushimi wrote:
nachomancer wrote:Consider the system ZFC+Axiom X, where axiom X is something like "For all P, P and not P" (assume for the sake of argument that ZFC is consistent). Any standard proof in ZFC also works in this new system, and they will stay consistent as long as we don't use Axiom X. Obviously this system is inconsistent, but it is not true that all of its theorems are "meaningless".

That doesn't in anyway contradict what I said. Proofs in ZFC + Axiom X are meaningless. I can prove the moon is made of cheese in ZFC + Axiom X. Proofs in ZFC are not meaningless.

That deserves to be stated more carefully. Is "meaningless" really the best word for describing proofs in an inconsistent system?

I completely agree that as soon as you find one contradiction in a formal system, you know the system cannot be relied on as a method of generating true statements. I probably would have been more OK with the word "useless" than "meaningless" (though it could be argued that experimenting with proofs in inconsistent systems can be useful too).

Proofs in an inconsistent system can be more than "meaningless", in the sense that individual statements in the proof can still have meaning.

Anyway, I realize we may be mostly just quibbling over words here, and I hesitate to be too pedantic, but then again, this is a web forum where we're talking about the philosophy of mathematics.


If the opposite of a given "true" statement can also be "proven", in what sense is any statement meaningful?
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Re: Math is a language, not a science?

Postby nachomancer » Sun Jun 05, 2011 9:41 pm UTC

Kurushimi wrote:That doesn't in anyway contradict what I said. Proofs in ZFC + Axiom X are meaningless. I can prove the moon is made of cheese in ZFC + Axiom X. Proofs in ZFC are not meaningless.


eSOANEM wrote:If the opposite of a given "true" statement can also be "proven", in what sense is any statement meaningful?


It is meaningful in the sense that it may still be true once you've modified the system to be consistent, just like math done before ZFC is still meaningful, even though the system it was originally proved was inconsistent. I think that we are using meaningful in two different ways. Proofs in inconsistent systems obviously can mean something, as I can turn any proof in a consistent system into one in an inconsistent system simply by adding on extra axioms, and then not using them.

That is what I mean by a meaningful proof in an inconsistent system: one that is still true in a consistent "subsystem" (whatever that is). I think this is a reasonable definition of meaningful, as it assigns the same label to the same object (a theorem) when you trivially modify the axiomatic system it was proved in, and it agrees with the normal definition of meaningful for standard proofs done in ZFC. If you are using a definition of meaningful that doesn't have this first property, than that would imply that if I were to find every book on foundations and pencil in Axiom X at the bottom of every list of the axioms of ZFC, than the entirety of mathematics would cease to be meaningful, or more realistically, that the discovery of Russell's Paradox made the statement 1+1=2 meaningless.

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Re: Math is a language, not a science?

Postby skullturf » Sun Jun 05, 2011 11:38 pm UTC

Yeah, I think different people seem to be using the words "meaningful"/"meaningless" in different ways.

If in the same conversation, or the same formal system, I say both "Harold is five feet tall" and "Harold is six feet tall", my statements contradict each other, but they are still meaningful.

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Re: Math is a language, not a science?

Postby Kurushimi » Mon Jun 06, 2011 1:37 am UTC

nachomancer wrote:
Kurushimi wrote:That doesn't in anyway contradict what I said. Proofs in ZFC + Axiom X are meaningless. I can prove the moon is made of cheese in ZFC + Axiom X. Proofs in ZFC are not meaningless.


eSOANEM wrote:If the opposite of a given "true" statement can also be "proven", in what sense is any statement meaningful?


It is meaningful in the sense that it may still be true once you've modified the system to be consistent,


That's my whole point. The meaningfulness is entirely contingent on the existence of a consistent system that can prove it. And if we can't find a consistent system to prove statements, then, yes, I would find the entirety of mathematics to be meaningless.

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Re: Math is a language, not a science?

Postby undecim » Mon Jun 06, 2011 1:54 am UTC

Mathematics can take many different forms. If we were to meet an alien civilization, they would certainly have different mathematical notation, but arrive at many of the same conclusions that we have.

That said, calling math a language is like calling written communication a language.
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Re: Math is a language, not a science?

Postby eSOANEM » Mon Jun 06, 2011 4:43 pm UTC

nachomancer wrote:
Kurushimi wrote:That doesn't in anyway contradict what I said. Proofs in ZFC + Axiom X are meaningless. I can prove the moon is made of cheese in ZFC + Axiom X. Proofs in ZFC are not meaningless.


eSOANEM wrote:If the opposite of a given "true" statement can also be "proven", in what sense is any statement meaningful?


It is meaningful in the sense that it may still be true once you've modified the system to be consistent, just like math done before ZFC is still meaningful, even though the system it was originally proved was inconsistent. I think that we are using meaningful in two different ways. Proofs in inconsistent systems obviously can mean something, as I can turn any proof in a consistent system into one in an inconsistent system simply by adding on extra axioms, and then not using them.


No. That statement is meaningful within the consistent system. In the inconsistent system, its opposite is just as "true" as it is and so it is meaningless to say one is "proven" within this inconsistent system.
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Re: Math is a language, not a science?

Postby skullturf » Mon Jun 06, 2011 5:15 pm UTC

eSOANEM wrote:No. That statement is meaningful within the consistent system. In the inconsistent system, its opposite is just as "true" as it is and so it is meaningless to say one is "proven" within this inconsistent system.


Well, even in an inconsistent system, you have proofs in the sense that you have derivations from the axioms.

In a system where you can prove both "P" and "Not P", both statements have been "proven" in the sense that they have been derived using axioms and rules of inference, but of course they haven't been "proven" in the sense of "shown to be actually true".


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