Me too.

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]]>Regardless our discussion, I hope the mathematician gets elected. For the sake of science!

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]]>http://www.stolaf.edu/people/steen/Papers/integrating.html

Integrating School Science and Mathematics: Fad or Folly?

Lynn Arthur Steen, St. Olaf College

“These differences are essential to proper understanding of a fundamental difference between mathematics and science:

Science seeks to understand nature.

Mathematics reveals order and pattern.

The subject of science is nature; the subject of mathematics is pattern. Each can contribute to the other, but they are fundamentally different enterprises. “

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]]>Again, not relevant. What anyone’s inspiration is has nothing to do with how their ideas are validated.

Let me again point out that by “observation” I am including measurement. In science, ideas are tested against the real world. From my earlier post “By establishing suitable postulates it is always possible to make a system of mathematics, just as Euclid did, but we cannot make a mathematics of the world, because sooner or later we have to find out whether the axioms are valid for the objects of nature.”

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]]>I would say we still have no idea what the other is talking about.

It is impossible for me to give such an example because math results cannot be disproven by observation. My whole point that math results are proved by recourse to assumptions, not observations.

To prove Pythagoras’ theorem, you do not measure anything. From wikipedia “Euclidean geometry … consists of assuming a small set of intuitively appealing axioms, and then proving many other propositions (theorems) from those axioms.” It is an entirely self-contained field.

The Pythagorean theorem is true in Euclidean geometry, but not in other types in which his 5th axiom is changed. From exampleproblems.com:

The Pythagorean theorem in non-Euclidean geometry

The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, the Euclidean form of the Pythagorean theorem given above does not hold in non-Euclidean geometry. For example, in spherical geometry, all three sides of the right triangle bounding an octant of the unit sphere have length equal to π / 2; this violates the Euclidean Pythagorean theorem because (\pi/2)^2+(\pi/2)^2\neq (\pi/2)^2.

This means that in non-Euclidean geometry, the Pythagorean theorem must necessarily take a different form from the Euclidean theorem. There are two cases to consider — spherical geometry and hyperbolic plane geometry; in each case, as in the Euclidean case, the result follows from the appropriate law of cosines:

* For any right triangle on a sphere of radius R, the Pythagorean theorem takes the form

\cos \left(\frac{c}{R}\right)=\cos \left(\frac{a}{R}\right)\,\cos \left(\frac{b}{R}\right).

By using the Maclaurin series for the cosine function, it can be shown that as the radius R approaches infinity, the spherical form of the Pythagorean theorem approaches the Euclidean form.

* For any triangle in the hyperbolic plane (with Gaussian curvature −1), the Pythagorean theorem takes the form

\cosh c=\cosh a\,\cosh b

where cosh is the hyperbolic cosine. By using the Maclaurin series for this function, it can be shown that as a hyperbolic triangle becomes very small (i.e., as a, b, and c all approach zero), the hyperbolic form of the Pythagorean theorem approaches the Euclidean form.

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]]>Now you give me an example of a mathematical theorem still used that has been proven wrong by observation. I’m not asking for one that hasn’t been yet proved because our powers of observation aren’t that well developed yet.

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]]>Or perhaps a certain higgins shouldn’t make silly assumptions.

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]]>Can you give me an example? [By observation I mean measurement and comparison with the world.]

I gave an earlier example of the Pythagorean example and what I thought were “scientific” and mathematical approaches to the problem. Also the text I scanned in says there is a difference between science and math.

[I see NK has bailed out.]

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]]>j a higginbotham — your premise is that mathematics is not a science because it is not based on observation. My premise is that mathematics is based on — and proven — by observation. Therefore it is a science, albeit one approached from another perspective.

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]]>No. Mathematics is proven based on its postulates. Below I have appended the Contents of The Annals of Mathematics, Vol. 169, No. 2, 2009. I looked at the abstracts but not the papers. I see nothing that suggests any of these papers are based on observance of nature. [Am I correct in assuming that you believe observance of nature to be a necessary component of science?]

DB: How else do mathematics make sense?

I don’t understand this question.

DB: If mathematics is separate from nature, surely I could prove that 4+3=8, could I not?

[If English is separate from nature, surely I could prove that “They is …” is proper usage, could I not?]

In math, statements can be proven. In science, statements are not “proved”. See the text I scanned in earlier.

DB: If 4+3 actually did equal 8, all of science would be sorta screwed.

—–

Why? Because there is some inconsistency? What about radiation – does light behave like particles or waves? Isn’t that a similar inconsistency?

http://annals.princeton.edu/annals/2009/169-2/index.xhtml

An embedded genus-one helicoid 347-448

Matthias Weber & David Hoffman & Michael Wolf

Nontangential limits in Pt(µ)-spaces and the index of invariant subgroups 449-490

Alexandru Aleman & Stefan Richter & Carl Sundberg

The geometry of fronts 491-529

Kentaro Saji & Masaaki Umehara & Kotaro Yamada

Curvature of vector bundles associated to holomorphic fibrations 531-560

Bo Berndtsson

The quasi-additivity law in conformal geometry

561-593

Jeremy Kahn & Mikhail Lyubich

Inverse Littlewood–Offord theorems and the condition number of random discrete matrices 595-632

Terence Tao & Van H. Vu

A combinatorial description of knot Floer homology 633-660

Ciprian Manolescu & Peter S. Ozsváth & Sucharit Sarkar

The concept of duality in convex analysis, and the characterization of the Legendre transform 661-674

Shiri Artstein-Avidan & Vitali Milman

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