You lied to us (about universal truths in math)
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 Viva El Shrooms
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You lied to us (about universal truths in math)
You lied to us in comic 263. Maths does not teach us perfect universal truths. My proof is thus. During Gaus' research into Euclidian geometry Gaus discovered that, mathmatically, no theory, a series of axioms, is consistant, all given axioms produce a correct answer, and consistant, all possible axioms are used. This includes number theory
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 NathanielJ
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Re: You lied to us
I believe you're referring to Godel's (not Gauss's) Incompleteness Theorem, one of the most popular and misunderstood theorem of mathematics.
But I don't see the problem. Aren't you in your very post taking that incompleteness theorem as a universal truth, thus proving the comic right?
But I don't see the problem. Aren't you in your very post taking that incompleteness theorem as a universal truth, thus proving the comic right?
Re: You lied to us
You're both misquoting and misunderstanding the Second incompleteness theorem. I would advise you to read its statement more closely to try and understand it better.
 NathanielJ
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Re: You lied to us
t0rajir0u wrote:You're both misquoting and misunderstanding the Second incompleteness theorem. I would advise you to read its statement more closely to try and understand it better.
If you're referring to the second half of my post, it was meant in jest.
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Re: You lied to us
NathanielJ wrote:t0rajir0u wrote:You're both misquoting and misunderstanding the Second incompleteness theorem. I would advise you to read its statement more closely to try and understand it better.
If you're referring to the second half of my post, it was meant in jest.
I think he meant that shrooms is both misquoting and misunderstanding, not that both of you are misunderstanding. I.e. the "both" refers to misquoting and misunderstanding, not to you and shrooms.
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 Viva El Shrooms
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Re: You lied to us
I am not misunderstanding or misquoting, gaus' work on Euclidian geometry is in itself either incompeate or inconsistant, as are all mathmatical theorys, I am not refering to Godel, whist his work was very good, Gaus was first, although unpublished his work on Euclidian geometry has been authenticated. I believe that you should refer youself to Euclids five postulates on geometry.
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Re: You lied to us
Now I am VERY confused about what he is trying to say. Gauss did nonEuclidean geometry, and as far as I know, there's nothing inconsistent about it, even if it is slightly counterintuitive. It's just a different geometry than Euclid's, and there is nothing wrong with it, even if it is incomplete. (does geometry follow number theory?)
And yes, mathematics does not teach universal truths, it universal truths with the assumption of a few basic axioms. If you believe that the axioms hold in this world, though, the mathematics does teach universal truths.
P.S. NonEuclidean geometry and Euclidean geometry are related. For example, projective nspace is just the normal (n+1)space modding out scalars.
And yes, mathematics does not teach universal truths, it universal truths with the assumption of a few basic axioms. If you believe that the axioms hold in this world, though, the mathematics does teach universal truths.
P.S. NonEuclidean geometry and Euclidean geometry are related. For example, projective nspace is just the normal (n+1)space modding out scalars.
 Viva El Shrooms
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Re: You lied to us
Maths is not just universal truths with axioms. i think you are misunderstanding what i said in my first post, axioms are arguments which are either true or false depending on what you apply them to. What you are thinking of are postulates, things which are assumed to be true, but cannot be proved, Euclids postulates on geometry are a very good example of this
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Re: You lied to us
I think you should get a better grasp of the English language (and communication in general) before telling others what certain words mean.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?
 Viva El Shrooms
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Re: You lied to us
what do you mean by that, i understand that mathmatical language is sometimes confusing, it took me a while to grasp it and i still have some dificulty when approaching new topics in maths
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Re: You lied to us
Viva El Shrooms wrote:axioms are arguments which are either true or false depending on what you apply them to. What you are thinking of are postulates
Axioms are the same thing as postulates. I think you should probably take the time to understand the foundations of mathematics a little more thoroughly before you take up this argument again.
And if you really understand that mathematical language is confusing, you should attempt to appreciate that the incompleteness theorems are more subtle than you think they are. Think of them along the same lines as the laws of thermodynamics  they constrain what we can do, but not to the point that we stop doing it. (Just because we can't make 100% efficient engines doesn't mean the car industry shuts down.)
 Viva El Shrooms
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Re: You lied to us
you have been reading to much wikipedia, but even there it is clear that they are only the same in traditional, logical arguments. In maths axioms are arguments, such as a^2b^2 does not equal (ab)^2, to be proved to support a theory. Whereas a postulate is a unproveable fact, such as you can draw a straight line between two points, used to decide whether an axiom is true or false.
This is beside the point, there are no universal thruth in mathmatics that can be proved without postulates which was my origanal point, and I am not saying that we cannot use mathsbecause of it, i was mearly pointing out the fact that the comic was wrong in an atempt to get people to debate about incompleateness in mathmatics, thankyou for paricipating
This is beside the point, there are no universal thruth in mathmatics that can be proved without postulates which was my origanal point, and I am not saying that we cannot use mathsbecause of it, i was mearly pointing out the fact that the comic was wrong in an atempt to get people to debate about incompleateness in mathmatics, thankyou for paricipating
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 Xanthir
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Re: You lied to us
In simple terms, you have that exactly backwards. In more correct terms, you're completely wrong.
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 Viva El Shrooms
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Re: You lied to us
Xanthir wrote:In simple terms, you have that exactly backwards. In more correct terms, you're completely wrong.
please dont just say that someone is wrong, explain how they are wrong, correct them and be pleasant about it.
so what do you think a postulate and an axiom is. dont refer to wikipedia, its rubbish
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Re: You lied to us
You might want to be a bit less presumptuous. There are a lot of people on these boards with extensive mathematical training and experience.Viva El Shrooms wrote:you have been reading to much wikipedia
No. An axiom is a given statement from which theorems are derived. In this case, the axioms would be those of symbolic algebra or whatever, and the theorem would be "a^2b^2 does not equal (ab)^2".In maths axioms are arguments, such as a^2b^2 does not equal (ab)^2, to be proved to support a theory.
Sure, but that's like saying "there is no painting which can be painted without paint". Mathematical truths are only relevant within the context of their axiomatic system, but that doesn't make them any less universal. For exampe, it's a universal truth that, given the axioms of number theory, there are infinitely many primes.there are no universal thruth in mathmatics that can be proved without postulates
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Re: You lied to us
first of all thank you for being polite, in response;
to your first statement, fair enough, i overstepped the mark, it simply seamed that the replies on this topic were informed by wikipedia.
to your second statement, i am not sure that i agree with the first half, but the latter half fair enough.
to your third statement, touche, but even number theory is incmopleate
to your first statement, fair enough, i overstepped the mark, it simply seamed that the replies on this topic were informed by wikipedia.
to your second statement, i am not sure that i agree with the first half, but the latter half fair enough.
to your third statement, touche, but even number theory is incmopleate
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Re: You lied to us
Viva El Shrooms wrote:first of all thank you for being polite, in response;
to your first statement, fair enough, i overstepped the mark, it simply seamed that the replies on this topic were informed by wikipedia.
to your second statement, i am not sure that i agree with the first half, but the latter half fair enough.
to your third statement, touche, but even number theory is incmopleate
Just because something is incomplete doesn't imply that it's poorlydefined or made of lies. To build on an earlier example: we don't know all the prime numbers, but we do know many of their properties without knowing them all. Some of them are proven and some of them are not, but as long as we can generalize the statements made about them to apply to all known primes, we have a very good case for stating its truth.
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Re: You lied to us
Viva El Shrooms wrote:to your first statement, fair enough, i overstepped the mark, it simply seamed that the replies on this topic were informed by wikipedia.
Perhaps the similarity is due to the mathematical content on wikipedia also being mostly written by people with extensive mathematical training? I've found the advanced math and physics entries there to be quite excellent. And I will add to the chorus that you are not using the word axiom correctly. Perhaps you're mixing it up with a lemma?
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 Xanthir
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Re: You lied to us
From what I can tell, he's mixing it up with "hypothesis". That is, something you assume, and then prove to be correct or not.
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 Viva El Shrooms
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Re: You lied to us
Bassoon wrote:Just because something is incomplete doesn't imply that it's poorlydefined or made of lies. To build on an earlier example: we don't know all the prime numbers, but we do know many of their properties without knowing them all. Some of them are proven and some of them are not, but as long as we can generalize the statements made about them to apply to all known primes, we have a very good case for stating its truth.
read the comic and my posts, i am not saying that maths is poorlydefined or made of lies, i am only saying that maths is incomplete, can we all agree on that
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Re: You lied to us
The incompleteness of number theory doesn't mean that theorems in number theory are not universal truths. In fact, every theorem in number theory tells us some truth about the natural numbers. Incompleteness just means that we can never know all of them.
I'm also not sure what you mean by geometry being incomplete; Tarski actually axiomatized Euclidean geometry as a firstorder theory, and in fact proved it to be complete. What is true is that the parallel postulate doesn't follow from the other postulates of Euclid, since there are nonEuclidean geometries in which the parallel postulate doesn't hold but the others do. So in that sense, geometry without the parallel postulate is incomplete, but that just means there are lots of different geometries.
I suspect this is why you are referring to Gauss, since he claimed to have discovered nonEuclidean geometry. However, he is not the person who first introduced it to the world, so other people generally get the credit.
I'm also not sure what you mean by geometry being incomplete; Tarski actually axiomatized Euclidean geometry as a firstorder theory, and in fact proved it to be complete. What is true is that the parallel postulate doesn't follow from the other postulates of Euclid, since there are nonEuclidean geometries in which the parallel postulate doesn't hold but the others do. So in that sense, geometry without the parallel postulate is incomplete, but that just means there are lots of different geometries.
I suspect this is why you are referring to Gauss, since he claimed to have discovered nonEuclidean geometry. However, he is not the person who first introduced it to the world, so other people generally get the credit.
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Re: You lied to us
To play the devil's advocate a bit, there's also the other incompleteness theorem, which implies that ZFC can't prove itself to be consistent.
So in some sense, if you don't have complete trust that the ZFC axioms^{*} are consistent, you shouldn't have complete trust in some statements in number theory either. Maybe you could argue from there that it wouldn't be correct to call them "universal truths".
If you think about it, though, Godel's theorem doesn't have anything to do with this. A proof of the consistency of ZFC within ZFC would be pretty pointless. Naive set theory can prove itself consistent just fine, precisely because it's inconsistent. Dig down far enough, and the whole thing boils down to "you can't start with no assumptions and derive useful knowledge", which is a pretty vacuous statement. Though maybe Descartes would disagree.
Besides, if tomorrow somebody discovered that ZFC actually was inconsistent, there'd be a some buzz about it, but, hey, 10 or 20 years later, mathematicians have patched it up with another set of axioms^{*}, and the math curriculum for anyone who isn't planning on doing math research is pretty much identical. Math didn't even have a foundation a century or two ago, and (a+b)(ab)=a^{2}b^{2} was still a universal truth.
^{*} "postulates", if you are both stupid and stubborn.
So in some sense, if you don't have complete trust that the ZFC axioms^{*} are consistent, you shouldn't have complete trust in some statements in number theory either. Maybe you could argue from there that it wouldn't be correct to call them "universal truths".
If you think about it, though, Godel's theorem doesn't have anything to do with this. A proof of the consistency of ZFC within ZFC would be pretty pointless. Naive set theory can prove itself consistent just fine, precisely because it's inconsistent. Dig down far enough, and the whole thing boils down to "you can't start with no assumptions and derive useful knowledge", which is a pretty vacuous statement. Though maybe Descartes would disagree.
Besides, if tomorrow somebody discovered that ZFC actually was inconsistent, there'd be a some buzz about it, but, hey, 10 or 20 years later, mathematicians have patched it up with another set of axioms^{*}, and the math curriculum for anyone who isn't planning on doing math research is pretty much identical. Math didn't even have a foundation a century or two ago, and (a+b)(ab)=a^{2}b^{2} was still a universal truth.
^{*} "postulates", if you are both stupid and stubborn.
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Re: You lied to us
Wikipedia is consistent with mathworld for the most part, so I would argue otherwise. In fact, it's rather surprising that it's more accurate and understandable than many text books. For one thing, it has much less typos, so it's something I check when my text books are making no sense.Viva El Shrooms wrote:please dont just say that someone is wrong, explain how they are wrong, correct them and be pleasant about it.
so what do you think a postulate and an axiom is. dont refer to wikipedia, its rubbish
Ah, I didn't know that. Should stick that in my knowledge bank somewhere.skeptical scientist wrote:I'm also not sure what you mean by geometry being incomplete; Tarski actually axiomatized Euclidean geometry as a firstorder theory, and in fact proved it to be complete.
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Re: You lied to us
On the Wikipedia side discussion:
Shroom: you started the post with a broad and strong claim, made with both spelling errors and arrogance. That is a form of trolling.
If there is universal truth, mathematics intersects it. If you define universal truth away in a precise sense... then you probably will do it with mathematics.
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Shroom: you started the post with a broad and strong claim, made with both spelling errors and arrogance. That is a form of trolling.
If there is universal truth, mathematics intersects it. If you define universal truth away in a precise sense... then you probably will do it with mathematics.
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 gmalivuk
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Re: You lied to us
Yakk wrote:That is a form of trolling.
I'm Greg Malivuk and I approve of this statement.
Shrooms: Calm down and cut it out with claiming what we're saying is rubbish. Also, dare I say, maybe lay off the shrooms as a source for your philosophical 'insights'?
This thread and your post in the LHC thread demonstrate that you don't actually have anywhere close to the understanding of these topics as a number of people on this board, who are trying their best to be helpful to you. If you're actually interested in learning and having your questions answered, it would behoove you to be a bit nicer about it. Also, as your location suggests you're a native English speaker, maybe give your posts a quick onceover for spelling and grammar errors before posting, as that would make it a lot easier for us to understand what you're asking...
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Re: You lied to us
Viva El Shrooms wrote:to your third statement, touche, but even number theory is incmopleate
Er, well, he forgot to demand the consistency of the axioms of number theory, but one could consider this included in the demand for truth of the axioms unless stated otherwise. But given that, he is absolutely undeniably right, it's logically fucking true.
And incmopleate? For the love of cute kittens, please!
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Re: You lied to us (about universal truths in math)
I'll assume the axiom thing is settled. If not, I"ll toss in my assurance that an "axiom" is something assumed to be true that is not provenlike "~Ea:Sa=0" or what have you. (I know the E is backwards, but I am too lazy to look up the TeX now.)
Gauss's work on nonEuclidean geometry did not lessen the "truth" of Euclidean geometry. Using the definitions of "point", "line", and "plane" as well as the Euclidean parallel postulate will still yield all of traditional geometry. It's not supposed to be "this is true for every conceivable set of mathematics you can think of."
Similarly, in number theory, Godel's Incompleteness Theorem does indeed state that it is necessarily incomplete. But just because we'll never be able to show that when you do thisandthat with this string's Godel number you get a theorem, does not mean that the things that are derivable in number theory are any less true. a + b = b +a in these axioms regardless.
In addition, it's true that no set of axioms can assert its own consistency (except, if I remember, those that are actually inconsistent). But since it can be shown in symbolic logic that "From a contradiction, anything follows," we know it is very unlikely for inconsistency to be a problembecause then, anything would be provable in the system.
So, yes, math's truths are still universal. But you must always be aware of context. Instead of thinking of math as "this is true" think of it more as "this is true, given this". This isn't too much of a leap, really. Think about physics. You will probably be told that W = F * D....given constant force parallel to displacement. The equation is still true, but there are always assumptions about it. If I just give you a piece of paper that says "a^{2} + b^{2} = c^{2}", and asked if it was true, what would you say? If you said yes, I could tell you that a=2, b=3, and c=7. To really know what's going on, you need to know what it's about, you need the context. You need me to say "...where a, b, and c, are the lengths of the legs and hypotenuse of a right triangle respectively."
I just got done reading GEB, so hopefully it's all fresh in my mind enough to be right.
Gauss's work on nonEuclidean geometry did not lessen the "truth" of Euclidean geometry. Using the definitions of "point", "line", and "plane" as well as the Euclidean parallel postulate will still yield all of traditional geometry. It's not supposed to be "this is true for every conceivable set of mathematics you can think of."
Similarly, in number theory, Godel's Incompleteness Theorem does indeed state that it is necessarily incomplete. But just because we'll never be able to show that when you do thisandthat with this string's Godel number you get a theorem, does not mean that the things that are derivable in number theory are any less true. a + b = b +a in these axioms regardless.
In addition, it's true that no set of axioms can assert its own consistency (except, if I remember, those that are actually inconsistent). But since it can be shown in symbolic logic that "From a contradiction, anything follows," we know it is very unlikely for inconsistency to be a problembecause then, anything would be provable in the system.
So, yes, math's truths are still universal. But you must always be aware of context. Instead of thinking of math as "this is true" think of it more as "this is true, given this". This isn't too much of a leap, really. Think about physics. You will probably be told that W = F * D....given constant force parallel to displacement. The equation is still true, but there are always assumptions about it. If I just give you a piece of paper that says "a^{2} + b^{2} = c^{2}", and asked if it was true, what would you say? If you said yes, I could tell you that a=2, b=3, and c=7. To really know what's going on, you need to know what it's about, you need the context. You need me to say "...where a, b, and c, are the lengths of the legs and hypotenuse of a right triangle respectively."
I just got done reading GEB, so hopefully it's all fresh in my mind enough to be right.
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Re: You lied to us (about universal truths in math)
In the WentworthSmith series on geometry, it is asserted that axioms differ from postulates in that axioms must be accepted as general unprovable truths, while postulates must be accepted as unprovable truths within a particular branch of mathematics. Now, these books are very old, and I haven't studied set theory in depth, but am I to understand from the preceding discussion that the concept of a postulate qua postulate has been subsumed by the concept of an axiom? That is, are there no more postulates "peculiar" to different branches of mathematics?
Tangentially, I've come to believe that it is a very grave pedagogic error to attempt to teach a branch of mathematics as just another extension of set theory. In my studies of the theory of computation, graph theory, and others, the single greatest hindrance to understanding has been that all concepts are defined in terms of sets. I've often had to look around online to find the way that a concept was defined before that definition was replaced by a set theoretic definition. Once I understand the "less rigorous" definition, I can easily see how it maps onto the set theoretic definition. But why in God's name do textbooks attempt to teach these concepts without explaining how they came about, how the people who came up with them understood them before everything became an application of set theory? Too many definitions are of the form, "An X is a set of subsets whose members...blah blah blah." Sure, the formal definition is more precise than an informal one, but the informal definition motivates the formal definitionnot vice versa.
Just as any textbook worth its salt explains why proofs work, textbooks should explain what formal definitions are really getting at and what inspired them.
Sorry, that was a bit of a rant.
Tangentially, I've come to believe that it is a very grave pedagogic error to attempt to teach a branch of mathematics as just another extension of set theory. In my studies of the theory of computation, graph theory, and others, the single greatest hindrance to understanding has been that all concepts are defined in terms of sets. I've often had to look around online to find the way that a concept was defined before that definition was replaced by a set theoretic definition. Once I understand the "less rigorous" definition, I can easily see how it maps onto the set theoretic definition. But why in God's name do textbooks attempt to teach these concepts without explaining how they came about, how the people who came up with them understood them before everything became an application of set theory? Too many definitions are of the form, "An X is a set of subsets whose members...blah blah blah." Sure, the formal definition is more precise than an informal one, but the informal definition motivates the formal definitionnot vice versa.
Just as any textbook worth its salt explains why proofs work, textbooks should explain what formal definitions are really getting at and what inspired them.
Sorry, that was a bit of a rant.
Re: You lied to us (about universal truths in math)
As far as I know, most graph theory text books comes with enough pictures that you can intuitively understand what it is talking about, even with the set notation they use. They (at least the ones I have) immediately points out that the vertex sets are the points, while the edge sets are the lines.
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Re: You lied to us (about universal truths in math)
I never found set notation particularly difficult to understand. The problem for me is the reverse; I can see the informal definition, but it's hard to work with until I get into the nittygritty workings of a particular object. The set definition makes it clear exactly what I have to work with. I guess this is a matter of preference though.
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Re: You lied to us (about universal truths in math)
The Mad Scientist wrote:In the WentworthSmith series on geometry, it is asserted that axioms differ from postulates in that axioms must be accepted as general unprovable truths, while postulates must be accepted as unprovable truths within a particular branch of mathematics. Now, these books are very old, and I haven't studied set theory in depth, but am I to understand from the preceding discussion that the concept of a postulate qua postulate has been subsumed by the concept of an axiom? That is, are there no more postulates "peculiar" to different branches of mathematics?
This actually relates quite strongly to your other problem, in that the only statements that we need to assume these days are the axioms of set theory, as everything has been reduced to set theory. It might make fields harder to understand, but it simplifies the assumptions that are required.
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Re: You lied to us (about universal truths in math)
I THINK where this comes from is the experiment that Gauss allegedly did involving creating a very large triangle using towers of various heights (so as to negate the curvature of the Earth) and then tried to determine if the triangle had any defect (thus making the universe nonEuclidean). The alleged result of the experiment was that the triangle had defect within experimental error and so Gauss could not rule out that the universe was Euclidean. A larger triangle needed to be constructed.
Now, I don't know if this experiment ever actually took place, but it is a nice story.
Now, I don't know if this experiment ever actually took place, but it is a nice story.
Re: You lied to us (about universal truths in math)
The Mad Scientist wrote:I've often had to look around online to find the way that a concept was defined before that definition was replaced by a set theoretic definition. Once I understand the "less rigorous" definition, I can easily see how it maps onto the set theoretic definition. But why in God's name do textbooks attempt to teach these concepts without explaining how they came about, how the people who came up with them understood them before everything became an application of set theory? Too many definitions are of the form, "An X is a set of subsets whose members...blah blah blah." Sure, the formal definition is more precise than an informal one, but the informal definition motivates the formal definitionnot vice versa.
I think this is a very serious problem with the way most mathematics textbooks are written. Formal exposition tends to be favored over historical development and motivation, possibly for the sake of brevity (we had a discussion about "mathematical machismo" here recently). For example, the most intuitive examples of groups (at least from a nonabelian perspective) all arise as a consequence of group actions  they are processes, not things, which is usually glossed over in a formal presentation of the subject. (It is also not stressed how important this paradigm shift from looking at groups as processes  for example, the historical development of Galois theory as a theory of permutations  to groups as objects is.)
I suppose the balance between abstraction (a la Grothendieck) and history and motivation is careful.
Edit: On the other hand, in the interest of defending the abstract (although not necessarily settheoretic) point of view, one very good reason why we like to reduce things to axioms and a minimal set of defining qualities is to capture "essentials": what results can we prove that don't use all of the axioms? What generalizes when we drop one of them and what doesn't? (For example, certain basic results that are true of groups are still true when you drop inverses.) How does that clarify existing theory? All of these are questions you can't answer until you go past the examples and understand the flow from axioms to theorems.
This point of view is not strictly settheoretic: it is also related to model theory and reverse mathematics.
jestingrabbit wrote:everything has been reduced to set theory.
I don't really agree with this statement; see the above two examples. But I don't really know enough to say more, and this is, to a first approximation, close enough to being true.

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Re: You lied to us (about universal truths in math)
achan1058 wrote:As far as I know, most graph theory text books comes with enough pictures that you can intuitively understand what it is talking about, even with the set notation they use. They (at least the ones I have) immediately points out that the vertex sets are the points, while the edge sets are the lines.
Graph theory didn't have too many awkward definitions. The theory of computation had more, specifically in an old textbook I got from the library entitled Elements of the Theory of Computation. The early settheoretic definitions of finite automata served more to hinder than to aid understanding. It would have helped if the formal definitions had been motivated by less formal, more intuitive definitions. The definitions seem perfectly obvious in retrospect, but the seemed obtuse when I first encountered them. Perhaps I'm just slow?
qinwamascot wrote:I never found set notation particularly difficult to understand. The problem for me is the reverse; I can see the informal definition, but it's hard to work with until I get into the nittygritty workings of a particular object. The set definition makes it clear exactly what I have to work with. I guess this is a matter of preference though.
Of course you want a strict, formal definition, especially since without such definitions proofs become less certain. But I don't believe that they should be presented without explanation or motivation.
jestingrabbit wrote:This actually relates quite strongly to your other problem, in that the only statements that we need to assume these days are the axioms of set theory, as everything has been reduced to set theory. It might make fields harder to understand, but it simplifies the assumptions that are required.
This only highlights the absurdity of the way mathematics is currently taught: previous generations, who had the benefit of knowing those less elegant yet easier to understand definitions, have banished them from the classroom in favor of their "improved" definitions. Abstract mathematics is presented as if its meaning, usefulness, and beauty should be selfevident to anyone with half a brain. In the Abstract Algebra textbook on my bookshelf, the subject is "motivated" in a single paragraph, and the remainder of the first chapter consists of a crash course in set theory and a dozen definitions. There is no context, and the reader is expected simply to wait until later chapters to find out why the subject is actually interesting or useful. Surely Abstract Algebra wasn't invented one day by someone who sat down and wrote out a bunch of settheoretic definitions.
In essence, higherlevel math textbooks read like they're written for interested peers, not for students who may be skeptical as to why the subject is even worth studying.
t0raijir0u wrote:On the other hand, in the interest of defending the abstract (although not necessarily settheoretic) point of view, one very good reason why we like to reduce things to axioms and a minimal set of defining qualities is to capture "essentials": what results can we prove that don't use all of the axioms? What generalizes when we drop one of them and what doesn't? (For example, certain basic results that are true of groups are still true when you drop inverses.) How does that clarify existing theory? All of these are questions you can't answer until you go past the examples and understand the flow from axioms to theorems.
I don't believe that such considerations should dictate the layout of introductory textbooks. Think about it this way: if we have a branch of mathematics which has a handful of postulates (in addition to the axioms of set theory), and someone manages to eliminate all of the postulates, this is seen as a significant featwhich implies that deriving the intuitive postulates from set theory is a nontrivial activity. When you reduce the number of axioms in a formal system, its theorems almost always become harder to prove. But this means that they become harder to follow as well! So we have a group of people who are so proud of their accomplishment (eliminating postulates) that they force students to follow more difficult proofs than they themselves had to follow when learning the subject. Wouldn't it make more sense to present the subject first with the additional postulates and then later, after the student has acquired an intuition for the subject, show how these postulates can be removed?
I think that more higherlevel math courses should be taught like calculus and analysis are: first you are given an intuitive, easy to understand definition of a limit, and a bunch of results are given with informal proofs or in some cases simple plausibility arguments. The subject as a whole is motivated by its applications to physics, engineering, and a host of other disciplines. Care is taken to explain not only why concepts are introduced and what they really mean, but to put them in historical context and explain how they came about.
Imagine taking an analysis course without ever having taken a course in calculus. ("Epsilon? Delta? What's all this about? Why should I care?") That's how most abstract mathematics is presented, unfortunately: all "analysis" and no "calculus".
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Re: You lied to us (about universal truths in math)
The Mad Scientist wrote:Imagine taking an analysis course without ever having taken a course in calculus. ("Epsilon? Delta? What's all this about? Why should I care?") That's how most abstract mathematics is presented, unfortunately: all "analysis" and no "calculus".
That's exactly what I did. First year analysis was all about the epsilons and deltas, and I'd never come across them before.
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Re: You lied to us (about universal truths in math)
jestingrabbit wrote:This actually relates quite strongly to your other problem, in that the only statements that we need to assume these days are the axioms of set theory, as everything has been reduced to set theory. It might make fields harder to understand, but it simplifies the assumptions that are required.
Sort of. Sure, you can derive much of mathematics without individual axioms from each theory  but those axioms describe the kind of things in which the theory applies. They are less unproved truths, and more a description of "if these things are true, then $theory is true"  find any structure in which the axioms are true, and you get the $theory out.
In many cases, the axioms and theory are set up as a settheoretic structure in a way that is more precise than it needs to be for the $theory to follow from the axioms. It is set up for ease of notation and proof.
The Mad Scientist wrote:Tangentially, I've come to believe that it is a very grave pedagogic error to attempt to teach a branch of mathematics as just another extension of set theory. In my studies of the theory of computation, graph theory, and others, the single greatest hindrance to understanding has been that all concepts are defined in terms of sets. I've often had to look around online to find the way that a concept was defined before that definition was replaced by a set theoretic definition. Once I understand the "less rigorous" definition, I can easily see how it maps onto the set theoretic definition. But why in God's name do textbooks attempt to teach these concepts without explaining how they came about, how the people who came up with them understood them before everything became an application of set theory? Too many definitions are of the form, "An X is a set of subsets whose members...blah blah blah." Sure, the formal definition is more precise than an informal one, but the informal definition motivates the formal definitionnot vice versa.
Some people actually find those set theoretic definitions extremely illuminating and useful. There is an old joke in CS  if you don't get CS until you run into the 5tuple definitions of a FSM (states, starting, terminal, alphabet, transition function), then say "oh, so that is what it is", you might be suited to being a mathematician. Or words to that effect.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
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Re: You lied to us (about universal truths in math)
I'd say those things are also reduced to set theory, they just happen to also be studying mathematical statements. When you prove things in either of these fields, you are still working in ZFC. In fact, the compactness theorem of model theory requires the axiom of choice (or a weaker version of it).t0rajir0u wrote:All of these are questions you can't answer until you go past the examples and understand the flow from axioms to theorems.
This point of view is not strictly settheoretic: it is also related to model theory and reverse mathematics.jestingrabbit wrote:everything has been reduced to set theory.
I don't really agree with this statement; see the above two examples. But I don't really know enough to say more, and this is, to a first approximation, close enough to being true.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?
Re: You lied to us (about universal truths in math)
To quote Yang, of "YangMills Theory":
That's why you should start with theoretical physics versions of maths, all the ideas, none of the set theory
More seriously, I read this about Category theory: That it allows one to study mathematical structures in an axiomatic enviroment more suited to what is actually being studied, as opposed to pulverizing them into sets first. I think the image used was trying to study the architecture of a cathedral by taking it apart stone for stone... ah found it, naturally it was John Baez, the theoretical physicist turned mathematician
There exist only two kinds of modern mathematics books: ones which you cannot read beyond the first page and ones which you cannot read beyond the first sentence.
That's why you should start with theoretical physics versions of maths, all the ideas, none of the set theory
More seriously, I read this about Category theory: That it allows one to study mathematical structures in an axiomatic enviroment more suited to what is actually being studied, as opposed to pulverizing them into sets first. I think the image used was trying to study the architecture of a cathedral by taking it apart stone for stone... ah found it, naturally it was John Baez, the theoretical physicist turned mathematician
Category theory allows you to work on structures without the need first to pulverise them into set theoretic dust. To give an example from the field of architecture, when studying Notre Dame cathedral in Paris, you try to understand how the building relates to other cathedrals of the day, and then to earlier and later cathedrals, and other kinds of ecclesiastical building. What you don't do is begin by imagining it reduced to a pile of mineral fragments.
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Re: You lied to us (about universal truths in math)
This is why I love Spivak's Geometry. His second volume starts with a translation of a paper by Gauss and one by Reimann. If you don't dig back to the roots, it's not at all clear what is "curving" when you define curvature in terms of connections on fiber bundles or whatnot. And because so much notation and language changes, you could spend years studying math and still find Gauss's original work unintelligible  but this should not be! I'd say that is the test of the ultra formal / Bourbaki style approach  does it actually equip you to go back to the older approaches? Maybe it does for some people, but I would be lost if that's all I had.
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Re: You lied to us (about universal truths in math)
Ok I was wromg in my understanding of what I said, and thank you for explaining my misconceptions in mathmatics, but from what i can see the comic was still wrong, do you agree
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