Math111:Community Portal
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This is the community portal page for Math 111, History of Mathematics. It is broken down into eleven sections with readings listed in each one. To post a reading response, first be sure to log in. Click on the name of the assignment, creating a new page if necessary. Then go to "discussion" and click the + sign next to "edit." Under "Subject/headline", put in "Response: John Doe" (with your name instead of John Doe, of course) and add your paragraph or two.
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Contents |
[edit] Intro
One of the main points of discussion in this intro unit is all the work that mathematicians make their definitions do. Here are a pair of great sages commenting on the same issue, from another famous dialogue written by a mathematician:
- "That's a great deal to make one word mean," Alice said in a thoughtful tone.
- "When I make a word do a lot of work like that," said Humpty Dumpty, "I always pay it extra."
[edit] Readings
2.2 (regular polyhedra)
22.2 (V-E+F)
Lakatos 142-154 "The Deductivist versus the Heuristic Approach"
[edit] Algorithms
Here are some links ([1], [2]) on the famous YBC 7289, the Babylonian tablet from sometime in the -1800 to -1600 range, expressing the square root of two sexigesimally as 1; 24,51,10. That's accurate to within about 0.0000006.
[edit] Readings
1.1,2,5 (Pythagorean theorem)
3.1-4 (Greek number theory)
5.1-3 (Chinese number theory)
6.1,2,3,5 (solving the cubic)
Al-Khwarizmi's quadratic formula - PDF
[edit] Infinities
Compare these pairs of quotes.
God and the infinite:
- For the nature of God is incomprehensible; that is to say, we understand nothing of what he is, but only that he is; and therefore the attributes we give him, are not to tell one another, what he is, nor to signify our opinion of his nature, but our desire to honor him with such names as we conceive most honourable amongst ourselves. (Thomas Hobbes)
- We know that the infinite exists without knowing its nature, just as we know that it is untrue that numbers are finite. Thus it is true that there is an infinite number, but we do not know what it is. (Blaise Pascal)
Georg Cantor and his influence:
- Cantor is a charlatan; renegade; corrupter of youth. (Leopold Kronecker)
- No one shall expel us from the paradise that Cantor has created for us. (David Hilbert)
[edit] Readings
Chapter 4 (Greek infinity)
10.1,2 (infinite series)
Rotman, Ad Infinitum - PDF ... Here's a reading guide to help you with this challenging reading.
[edit] Axiomatization
- The facts of mathematics are verified and presented by the axiomatic method. One must guard, however, against confusing the presentation of mathematics with the content of mathematics. An axiomatic presentation of a mathematical fact differs from the fact that is being presented as medicine differs from food. It is true that this particular medicine is necessary to keep the mathematician at a safe distance from the self-delusions of the mind. Nonetheless, understanding mathematics means being able to forget the medicine and enjoy the food. (Giancarlo Rota)
- If Logic is the hygiene of the mathematician, it not his source of food; the great problems furnish the daily bread on which he thrives. (Andre Weil, of Bourbaki)
[edit] Readings
2.1 (Euclid's setup)
18.1,2 (parallel postulate)
21.7,8 (ring axioms and biographies)
23.2,5 (sets and diagonalization)
Elements, Principia, Elements - reading link
[edit] Symbols and Formulas
Alfred North Whitehead (of Russell and Whitehead fame):
- By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that ... a large proportion of the population of Western Europe could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility ... Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation. [...] By the aid of symbolism, we can make transitions in reasoning almost mechanically, by the eye, which otherwise would call into play the higher faculties of the brain. [...] It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle -- they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.
Fernando Gouvea, from Euler's Convincing Non-Proofs:
- Computing the coefficient of x in the power series, he gets... a famous result that had been obtained by Leibniz. And he comments
- From this result, if there were any doubts as to the method, the whole sky is lighted up, so that there can be no doubt as to the validity of the new results were are to derive from this method.
- In other words, "See? It works!"
- ... What shall we say, then? Are we wrong to insist on rigorous proofs? Is there a special category of argument, something less than full proofs, something more than blowing smoke? Or do truly great mathematicians get special dispensation?
[edit] Readings
6.6 (cubics)
9.4 (product formula)
10.4 (Euler's sum)
10.6 (generating functions)
Hardy, The Indian Mathematician Ramanujan - PDF
Dunham, Euler, The Master of Us All - PDF PDF
[edit] Numbers
- But since such a remainder is negative, you will have to imagine √-15 - that is the difference between AD and 4AB - which you add to or subtract from AC, and you will have that which you seek, namely 5 + √25 - 40 and 5 - √25 - 40, or 5 + √-15 and 5 - √-15. [Dismissis incruciantionibus], multiply 5 + √-15 by 5 - √-15, making 25-(-15) which is +15. Hence this product is 40. ... So progresses arithmetic subtlety the end of which... is as refined as it is useless.
(Cardano in Ars Magna)
- The colorful Latin phrase Cardano used for this is dismissis incruciantionibus, and the translator notes that Cardano might very well be playing on a possible double meaning of this phrase in the sentence, which can be read either as "Dismissing mental tortures, multiply 5 + √-15 by 5 - √-15..." or as "Cancelling out cross-multiples, multiply 5 + √-15 by 5 - √-15..."
(Barry Mazur in Imagining numbers, particularly the square root of minus fifteen)
[edit] Readings
14.1-5 (complex numbers)
Chapter 21 (algebraic number theory)
Rotman, Signifying nothing - PDF
[edit] Paper Proposals
SUBMIT paper proposals here on the wiki. Feel free to update them as you refine your paper ideas.
[edit] Space
[edit] Readings
Chapter 7 (analytic geometry)
15.2 (stereographic projection)
18.6,7 (C in non-Euclidean geometry)
19.5 (Erlangen)
22.8 (biography of Poincare)
Greenberg, Philosophical Implications of non-Euclidean geometry - PDF
Poincare's papers in Stillwell Sources - PDF
[edit] Genius
[edit] Readings
Tony Rothman, The fictionalization of Evariste Galois - PDF
6.7 (solvability of polynomials)
19.1-4,7 (groups and Galois)
Duchin, The sexual politics of genius - PDF (related: selection from Men of Mathematics PDF)
[edit] Method, Practice, Meaning
- I don't believe or disbelieve the Riemann Hypothesis. I have a certain amount of data and a certain amount of facts. These facts tell me definitely that the thing has not been settled. Until it's been settled it's a hypothesis, that's all. I would like the Riemann Hypothesis to be true, like any decent mathematician, because it's a thing of beauty, a thing of elegance, a thing that would simplify many proofs and so forth, but that's all.
(A. Ivic)
- It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp, holds the key to a variety of scientific and mathematical investigations. The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum.
(John Derbyshire, "Prime Obsession")
- It's remarkable how the Riemann zeta function seems to be trying intentionally to deceive us!
(Warren D. Smith, "Cruel and unusual behavior of the Riemann zeta function")
[edit] Readings
9.1-2 (early calculus)
10.7 (zeta)
15.5 (Riemann biography)
[edit] Pure Math
- The theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics. The accusation is one against which there is no valid defence; and it is never more just than when directed against the parts of the theory which are more particularly concerned with primes. A science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life. The theory of prime numbers satisfies no such criteria. Those who pursue it will, if they are wise, make no attempt to justify their interest in a subject so trivial and so remote, and will console themselves with the thought that the greatest mathematicians of all ages have found it in it a mysterious attraction impossible to resist.
(G. H. Hardy from a 1915 lecture on prime numbers)
[edit] Readings
REVIEW (you can post here a response on any reading or lecture content from the quarter)
