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It's been many years since I did any real mathematics but last night after pondering the process involved in my mathematical journey I had an idea about the abstraction of how mathematical analysis works.

As we know, mathematics is about transforming logical statements into different forms. We use a series of tautological statements to get from point A to point B and hence A and B are equivalent, logically speaking. B maybe be more appropriate/useful in some instance than A even though it is mathematically equivalent.

Therefore, suppose a sphere represents an equipotent surface where any path along the sphere represents a "equivalence" or tautological movement. That is, any two points on the sphere are tautologically equivalent.

Then the majority of pure mathematics can be seen as finding paths along the surface of the sphere. A "mathematical proof" would essentially be a closed path on the sphere.

Approximations could be seen as moving off the surface of the sphere(to a new sphere).

In fact, we would not necessarily have spheres(but this seems like it would be a topological space) but arbitrary surfaces and it would require a higher dimension than 3. (since we can approximate functions in multiple ways with each one not necessarily being equivalent yet still close to the original).

One issue of the above has is that there seems to be no real metric for "Closeness" although maybe something could be developed. e.g., given two equivalent mathematical functions or statements, say point A and B, then how close is A to B? This would be required to visualize such things in a metric space(which is initialize how I conceived of it but not necessarily how it is).

In any case, the question is about such higher meta-mathematical analysis of knowledge. The above applies to just about anything where one thing is transformed into something else through equivalence. Equivalence derivation "moves one element to another along the same "dimension"" and approximate derivations move normal to that dimension.

Is there any theories out there like this that I could read more about?

Archival
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    In the second paragraph, "get from point $A$ to point $B$" is the implication $A\implies B$ not the equivalence $A\iff B$ – Nameless Jan 03 '13 at 16:32
  • I think a good place to investigate this formally is mathematical logic – Amr Jan 03 '13 at 16:32
  • @Nameless: My thoughts exactly. And as we rather have $A\Rightarrow B$ we also can no longer speak about an "equipotent surface" – Thomas Jan 03 '13 at 16:42
  • @Nameless: no, the surface is an equivalence relation on tautological relationships, so it is vacuous that A <==>. That is, every point on the surface is equivalent or tautological(Either one will do). Points are mathematical relationships/concepts/ideas/functions that just "look" different. – Archival Jan 03 '13 at 17:05
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    @Archival You said that before the introduction to your surfaces. It is thus unclear whether you mean implication or equivalence. Please edit the question to clarify that – Nameless Jan 03 '13 at 17:08
  • @Nameless, I never bring up implication. But it does not matter because the mathematical process of "proof" is tautological. Each statement in a proof must be tautological to all others or the proof is invalid(except for approximations, which requires a sort of closeness that I kinda talk about). – Archival Jan 03 '13 at 17:14
  • Maybe by using the term surface is confusing. You have to realize it is not a surface because a surface has the concept of closeness.Tautological statements are identical abstractly(just different "forms". For example, a diamond has many facets that may look different but ultimately it is just a diamond. – Archival Jan 03 '13 at 17:16
  • Basically it boils down to saying that all equivalent mathematical concepts are the same thing, they just "look" different. We move between the same concepts because they feel different but ultimately they are the same abstract concept. This process of movement is interesting to me but no matter what must be tautological or we end up with an illogical movement. – Archival Jan 03 '13 at 17:20
  • There is a simple proof that $x>0 \Rightarrow x^2>0$, however it is not true that $x^2>0 \Rightarrow x>0$. How does this fit into your framework? You can't construct a "closed path" between these two statements. Does your theory simply not consider such proofs? Or are you saying that the two statements "lie on different spheres"? You may find it instructive to read about preorders and posets, which are the natural setting for proofs, and maybe about categorical logic – Chris Taylor Jan 03 '13 at 17:42
  • Thinking of proof as relying on tautology as opposed to implication is awkward. I can prove that $1>0$ using the Peano axioms and the relevant definition of $>$, but surely can't prove the Peano axioms from $1>0$. You can stick to tautologies by explicitly keeping the axioms around: $A \iff A \wedge X_1 \iff A \wedge X_1 \wedge X_2 \cdots$, where $A$ is the conjunction of the axioms and $X_i$ are derived facts, but clearly there's a notion of "downstream" here that is lost when you only observe that these are tautologies. – mjqxxxx Jan 03 '13 at 17:44
  • @ChrisTaylor Those are not tautological and hence do not lay on the same surface. I do not know how they work into the framework. Possibly through approximations but I haven't given it much though. I am considering "surfaces" that consist of tautological statements. Statements are not tautological and hence are not on the same surface. Your "path" goes from one surface to another. (sort of like approximations as I have mentioned) – Archival Jan 03 '13 at 17:55

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First off, I'm not altogether what you are saying, let alone what you are asking.

You argue:

As we know, mathematics is about transforming logical statements into different forms. We use a series of tautological statements to get from point A to point B and hence A and B are equivalent, logically speaking.

Correction/clarification: mathematics does use tautological statements to get from "point A to point B", but it also uses one directional inferences, e.g., the material conditional, to get from "point C to point D", and implications to get from D to E, hence we can infer there's a way to get from point C to E. One may not, however, "backtrack". $(C\rightarrow D \rightarrow E) \not\Longrightarrow (E \rightarrow D \rightarrow C).$ Also, $[P \land (P\rightarrow Q)] \rightarrow Q$, but we cannot then logically infer that $Q \rightarrow P$.

So much of the rest of your argument makes little sense, given your characterization of math as consisting strictly of tautologies.

It might help if you take a look at the previous math.SE post: Common Misperceptions About Math

It seems as though you are trying to model, say, the domain of mathematical knowledge, from within that domain by using a field/concept in a sub-domain (e.g., topological space) which emerged within the very domain you're trying to model.

What I'm trying to say that there seems to be something very "circular" about your thoughts, or analogies...

Though I am very very open to discussing the philosophy of math, perhaps more so, the philosophy of mathematical logic and practice, because I don't think enough mathematicians step "outside" of math to evaluate the assumptions they take as given. Extraordinary work has been done in the Philosophy of Science, in this respect. Mathematics is as much about the mathematicians doing math (their assumptions as to what we take as true, their conventions which are implicitly adopted or rejected, the degree to which they believe that the work they do can be construed as purely objective, the contexts in which theories emerge, and the people "doing the thinking")... as it is about their thoughts and the products of their work.

Perhaps we need to step out of math (or rather, step into "meta-math" - perhaps higher-order math/logic) to be able to say anything about mathematical knowledge, mathematical thought, mathematical logic, and mathematical practice, rather than trying to model math from within the very conceptual domains/domain which emerged from the very model they may be trying to model.

Book suggestions:

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amWhy
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  • Yes, ultimately all knowledge is circular, I suppose what I have said makes it explicit. All definitions are circular, mathematical proofs are circular(although we only need to go around part way since we use tautologies, which force any reasoning to be circular). – Archival Jan 03 '13 at 17:17
  • e.g., if A <==> B, B <==> C, C <==> D, then D <==> A which "completes" the loop. Tautology is transitive, of course. – Archival Jan 03 '13 at 17:18
  • I'd suggest that this loop does not necessarily admit of strictly double implications, and we may find that the chain never returns to A. – amWhy Jan 03 '13 at 17:22
  • It must if the process is tautological. In any case, the point is not that it may or may not because we call those that do a surface and those that don't lay on another surface. – Archival Jan 03 '13 at 17:22
  • So, for example, an approximation is not bidirectional(tautological) but, as I said, we are moving "normal" to the surface(but one that is close). – Archival Jan 03 '13 at 17:23
  • I don't think deduction is always tautological. – amWhy Jan 03 '13 at 17:23
  • Yes, derivation is generally more specific. I am not talking about that. I am talking about "identical" mathematical objects. Is x + 6 = 3 the same as x + 9 = 6? at least in the standard algebraic space... – Archival Jan 03 '13 at 17:27
  • Now, there are an infinite other number of equations that are identical, right? so essentially any of them will work. Basic algebra: x + 2 = 9, solve for x: x + 2 = 9, apply a tautological process: x + 2 = 9 <==> x + 1 = 8(sub 1 from both sides), again: x = 7. x is the same in all equations though. So, in some sense, all the equations are "equivalent". (or lay on the same surface) – Archival Jan 03 '13 at 17:29
  • If each equation is a point on a surface then our "solution" is a "movement" from one point to another. Someone else might take a different path or even a short cut. – Archival Jan 03 '13 at 17:31
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You might be interested in reading about "topological psychology", which has aspects very similar to what you're asking about. This approach seems to have begun with Kurt Lewin's 1936 book Principles Of Topological Psychology. I remember looking at this book sometimes when I was in the library back in the 1970s as an undergraduate, but I could never figure out whether there was anything quantifiably non-trivial in the book.

Googling psychology AND topology seems to suggest that others have gone down the same road that Lewin did. See, for example, Topological Foundations of Cognitive Science by Barry Smith.

Dave L. Renfro
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  • I think what I am talking about is more about the "flow"(or process" of knowledge and not so much psychology. Think of it like this. You start with x + 2 = 6 <==> x = 6 - 2 <==> x = 4 <==> x + 2 = 6. Each "step" is equivalent to the previous(which is why it is useful) and each object, mathematically is really the the same. They just look different. Mathematics is about manipulating abstract mathematical objects to view them from different perspectives. but each equation above could be thought of as a point on a surface, the surface itself being the abstract mathematical object representing – Archival Jan 03 '13 at 17:10
  • all the equivalent objects that look different. (in this case we have 3 an infinite number of equations). When you "solve" something you are going from one point to another, the new point is easier to understand, usually. The abstract process, though, is very simple finding tautological relationships on "form"(but the "function" is the same in all cases). – Archival Jan 03 '13 at 17:12
  • I think it is quite safe to say there isn't anything mathematically non-trivial about it; who knows if psychologists actually find this stuff useful. As far as I understand, when one reads things like "the phallus is the square root of negative one", or that the torus is "exactly the structure of the neurotic", these people are making (often not-fully-informed) analogies which are not intended to have any mathematical content. (So it's not quite as bad as Sokal and Bricmont make it out, though still terribly obscurantist in my opinion.) – Zev Chonoles Jan 03 '13 at 17:44
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    @Archival: If your goal is to keep your ideas mathematically contentful (such as for example these people's ideas), and not only to make a nice analogical story where your ideas about knowledge are described using math words, I think you would be well-advised to be careful. – Zev Chonoles Jan 03 '13 at 17:56

  • FWIW, Harary includes the book in the bibliography of his textbook on graph theory.

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