Biographies, philosophy of mathematics, mathematics education, recreational mathematics, communication of mathematics

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24
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8answers
2k views

Has philosophy ever clarified mathematics?

I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any ...
4
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0answers
280 views

What is the *smallest* number used in a mathematical paper? [on hold]

Numbers such as Skewe's Number and Graham's Number and a few others are relatively well-known in mathematical folklore as being among the largest, if not the largest, numbers encountered in proper ...
15
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7answers
4k views

Why have mathematicians used differential equations to model nature instead of difference equations

Ever since Newton invented Calculus, mathematicians have been using differential equations to model natural phenomena. And they have been very successful in doing such. Yet, they could have been just ...
5
votes
2answers
170 views

A. Markov's papers?

A. Markov published several papers on his chains, starting in 1906, so it is written, in the journal: (1) Извѣстія Физико-математического общества при Казанском университете I am surprised by the ...
11
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0answers
175 views

Who first showed that $SL(n,O_K)$ is a lattice for a number ring $O_K$?

Let $O_K$ be the ring of integers in an algebraic number field $K$. Assume that $K$ has $r$ real embeddings and $s$ pairs of complex conjugate complex embeddings. There is then an injective ...
31
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1answer
2k views

Did Leibniz really get the Leibniz rule wrong?

A couple of posts ([1], [2]) on matheducators.SE seem to suggest that Leibniz originally got the wrong form for the product rule, perhaps thinking that $(fg)'=f'g'$. Is there any actual historical ...
7
votes
1answer
654 views

Did differential geometry undergo a notation change?

As a graduate student, I found the old books of differential geometry used a different set of notation from modern textbooks. For example, Chern and Milnor defined the curvature 2-form by ...
3
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0answers
47 views

Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem

If $G\subsetneq\mathbb{C}$ is a simply-connected plane domain, then by the Riemann mapping theorem there is a conformal isomorphism $\newcommand{\D}{\mathbb{D}}\varphi:\D\to G$, where $\D$ is the unit ...
32
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1answer
2k views

Hilbert's Hotel

Hilbert's Hotel is a famous story about infinity attributed to David Hilbert (1862-1943). Is it documented that Hilbert's Hotel is in fact due to Hilbert, and if yes, where?
10
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1answer
238 views

History of powers beyond squares and cubes

The ancient Babylonians understood squares:       Plimpton 322 The ancient Athenians understood cubes, if we can take doubling the cube, i.e., the Delian problem, as evidence. My ...
6
votes
2answers
346 views

How did height in algeb. number theory/elliptic curves started?

Maybe this is obvious but it isn't to me yet. What is the history of heights used in say points of the project plane over a number field or of elliptic curve over a number field? I would guess people ...
1
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0answers
131 views

Motivating mathematics(particularly algebraic number theory) through historical problems [closed]

Most mathematical textbooks start a subject by going backwards, historically. They will define the terms that were invented to solve a problem in their polished form and then use these definitions and ...
11
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2answers
463 views

History of the analytic class number formula

The (general) analytic class number formula gives a value for the residue of the Dedekind zeta function of a number field at the point $s=1$ (or, as I prefer, the leading Taylor coefficient at $s=0$). ...
6
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1answer
317 views

Different approaches to forcing

There are many different approaches to the forcing method, and I am looking for all known such approaches. So my question is: Question 1. Which different approaches to set theoretic forcing are ...
6
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0answers
218 views

Silver's unpublished work on reverse Easton iteration

Silver was the first person who used the method of reverse Easton iterations in connection with large cardinals, and used it to force the failure of $GCH$ at some measurable cardinal. At most papers ...
20
votes
2answers
760 views

construction of nonmeasurable sets

I have a history question for which I've had trouble finding a good answer. The common story about nonmeasurable sets is that Vitali showed that one existed using the Axiom of Choice, and Lebesgue et ...
2
votes
1answer
333 views

Who defined and who coined “module”?

The title of my Q. says it all: QUESTION:   Who defined and who coined: module? Would it be Emmy Noether? EDIT   In view of @anon's and KConrad's answers, and as it could have been ...
11
votes
2answers
372 views

Who originated the standard symbols for Lie groups GL, SL, SU, etc.?

Who was first to use symbols GL, SL, O, SO, U, SU, Sp and their projective versions, and how did this notation become standard? The notation appears in fairly modern form in Weyl's "The Classical ...
1
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0answers
103 views

First Description of how to Remove Radicals from Equations

Who first described the technique of removing radicals as indicated in the answers to questions Tools for Removing Radicals from Equations and Rewrite sum of radicals equation as polynomial equation ? ...
4
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1answer
364 views

The ten martini problem - reason for name

Why is the problem called the ten martini problem? Sounds like an interesting name for people who drink.
6
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1answer
168 views

Finiteness as a motivation for compactness

Another history question, and I am not sure if I will get any answers. (If anyone knows of a good history of math list to use for this question I would be happy for any tips. The one I used to post to ...
11
votes
3answers
1k views

How were formulas / images added to books in post-printing-press / pre-digital times?

I have seen that Euclid's Elements was written 300 BC and first set in type in 1482. Are there scans of that old versions available? How were formulas / images added to the books created with ...
5
votes
2answers
649 views

Salvaging Leibnizian formalism?

Can one justify Leibniz's formalism in a suitable algebraic or topological context? We have published some papers recently where we argue that Leibniz's formalism for the calculus wasn't ...
29
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1answer
3k views

A topologist is not a mathematician - a small question

Years ago I read about a topologist who was to enter the states as an immigrant and was asked a question about his profession. He indicated he was a topologist, but as this was not included on the ...
2
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0answers
70 views

History of limit point compact -/-> compact example

A standard example in elementary topology (e.g. Munkres) of a space which is limit-point compact (every infinite subset of the space has a limit point) but not compact is the minimal uncountable ...
5
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1answer
108 views

Origins of the Jacobi matrix

I have several questions concerning history of Jacobi matrices. Does anybody know why the Jacobi matrix (=symmetric tridiagonal matrix) is named by Carl Gustav Jacob Jacobi? What was his contribution ...
1
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0answers
98 views

Default Orientation of Vectors [closed]

When I started studying math in 1982 in Germany, there seemed to have been a change in the choice of the default orientation of vectors; while it was row-vectors till then, it changed to ...
12
votes
2answers
584 views

Maximal ideals are prime (history answer please!)

Please can someone tell me the history of the simple argument that any maximal ideal of a commutative ring or distributive lattice is prime? (It is understood that we have found the maximal one using ...
10
votes
3answers
862 views

When did coordinate plane “as we know it” come into play?

This is a historical question that needs some background to make sense. Let me start with the longer version of the question: When did negative numbers, algebra and coordinate plane come together? ...
2
votes
2answers
116 views

Where can I find a translation of Caspar Wessel's “Om directionens analytiske betegning?”

I found a listing on Google books for a book containing the desired English translation, together with some biographical information on Wessel, and entitled On the Analytical Representation of ...
16
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2answers
681 views

History of the connection between Riemann surfaces and complex algebraic curves

As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...
12
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2answers
700 views

Did Cauchy think that uniform and pointwise convergence were equivalent?

I've heard that Cauchy thought he'd proved that pointwise and uniform convergence are equivalent. Is this a historical fact? If it is indeed true, I was wondering if anyone had a reference.
30
votes
6answers
2k views

Negative impact of wrong or non-rigorous proofs

The recent talks of Voevodsky (for example, http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2014_IAS.pdf), which describe subtle errors in proofs by him as well as others, as well ...
35
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16answers
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Rediscovery of lost mathematics

Archimedes (ca. 287-212BC) described what are now known as the 13 Archimedean solids in a lost work, later mentioned by Pappus. But it awaited Kepler (1619) for the 13 semiregular polyhedra to be ...
21
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0answers
464 views

History of the Proj construction in algebraic geometry

Projective geometry was introduced by fifteenth century Renaissance painters (like Alberti, da Vinci and Dürer) in the guise of perspective theory, although one could argue that Pappus was already ...
33
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5answers
3k views

What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do they tend to succeed or fail?

I think we all occasionally come across terminology that we'd like to see supplanted (e.g. by something more systematic). What I'd like to know is, under what circumstances is it reasonable to believe ...
27
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26answers
4k views

Mathematicians who made important contributions outside their own field? [closed]

It is often said that scientists who cross disciplinary borders can make unexpected discoveries thanks to their fresh view of the problems at hand. I am looking for mathematicians who did just that. ...
33
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2answers
4k views

How did “normal” come to mean “perpendicular”?

How and when did the word "normal" acquire this meaning? When I first thought of this, I couldn't really come up with any explanation that wasn't complete speculation -- pretty much all I was able to ...
10
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1answer
301 views

Elements of the method of forcing in some papers of N. N. Luzin

In the paper Eléments de la méthode de forcing dans quelques travaux de N. N. Lousin. (French) [Elements of the method of forcing in some papers of N. N. Luzin] Amphora, 469–479, Birkhäuser, ...
7
votes
2answers
469 views

Survey of the history of calculus?

Boyer 1939 is a nice readable survey of the history of the calculus, but it's showing its age. Discussing the notion of instantaneous velocity, he has: Mathematics knows no minimum interval of ...
9
votes
2answers
426 views

Origin of the term “generic” in set theory

In set theory, in particular the context of forcing, if $M$ is a model of $\sf ZFC$ and $P\in M$ is a partial order, we say that $G\subseteq P$ is a generic filter (or $M$-generic or generic over $M$) ...
25
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3answers
2k views

Who invented diagrammatic algebra?

There is a strong and growing trend to do mathematics via diagrammatic algebra, which involves constructing and manipulating equations whose elements are diagrams drawn in the plane. The manipulations ...
13
votes
1answer
601 views

Who first dubbed them “expander graphs”?

Expander graphs ("sparse graphs that have strong connectivity properties") burst onto the mathematical scene around the millennium, but I have not been successful in tracing the origin of (a) the ...
3
votes
1answer
371 views

Levi's book on Leibnizian calculus

Raphael Levi learned from Leibniz at a late stage in Leibniz's career. This might be a definite advantage for understanding Leibniz. Leibniz did not elaborate some of the philosophical principles ...
7
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0answers
121 views

Origin of Lie Product Formula

I'm interested in where Lie wrote down the Lie Product formula (for finite matrices) (the precursor of the Trotter product formula; see http://en.wikipedia.org/wiki/Lie_product_formula). With a ...
5
votes
1answer
359 views

What was Seifert's contribution to the Seifert-van Kampen theorem?

The Seifert-van Kampen theorem is the classical theorem of algebraic topology that the fundamental group functor $\pi_1$ preserves pushouts; more often than not this is referred to simply as the van ...
7
votes
1answer
779 views

Is there a “big program” in mathematics at the moment? [closed]

I apologize in the event that you should find this question off topic. Please feel free to delete it if that is the case. Years ago, I studied undergrad mathematics at university. The understanding ...
5
votes
1answer
233 views

When was the word “stable” first used to describe stable homotopy theory?

The word "stable" has many uses in mathematics, but in the context of stable homotopy theory, one might take it to mean one of two things: Homotopy groups stabilize after taking suspensions ...
36
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4answers
3k views

History of “without loss of generality”

"Without loss of generality" is a standard in the mathematical lexicon, and I am writing to ask if anyone knows where the expression was popularized. (The idea has been around since antiquity, I'm ...
11
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3answers
2k views

Why are smooth numbers called “smooth”?

"Adleman refers to integers which factor completely into small primes as “smooth” numbers." (ME Hellman, JM Reyneri. Advances in Cryptology, 1983: citation link.) Does anyone know what is the ...