stoutfellow: (Winter)
To accept Intuitionism is, among other things, to accept that some meaningful mathematical statements are neither true nor false. From the standpoint of symbolic logic, this entails introducing additional truth values beside True and False; in other words, to contemplate "multi-valued logics".

On the gripping hand )

Ramble Contents
stoutfellow: My summer look (Summer)
In our last episode, a couple of months ago, we discussed the Intuitionists, and in particular their view that mathematics takes place in the human mind and, therefore, must be founded and justified in mental terms. This has enormous consequences.

Goodbye, Mr. Boole )

Ramble Contents
stoutfellow: My summer look (Summer)
The Logicists attempted to establish the truth of mathematics by grounding it in indisputable logic. The Formalists avoided the question of mathematical truth altogether. The third major school, Intuitionism, led by the great Dutch topologist L. E. J. Brouwer, took another tack, by redefining "truth".

Everybody Knows! )

Ramble Contents
stoutfellow: My summer look (Summer)
The last axiom in Zermelo's list requires a post of its own. It is rather complicated to describe, and - unusually - it is quite controversial. Details, as usual, under the cut.

And You Takes Your Choice )

Ramble Contents
stoutfellow: My summer look (Summer)
Both the Logicist and Formalist factions had an interest in establishing a set of axioms for set theory; the paradoxes that Cantor's set theory led to were unsatisfying to both. Several sets of axioms have been suggested; I'm going to describe one of the first, devised by Ernst Zermelo in 1908, under the cut.

What is a set? )

Ramble Contents
stoutfellow: My summer look (Summer)
The Formalist program required the selection of a list of fundamental axioms from which all of mathematics could be (in principle) constructed. More broadly, each branch of mathematics should have its own list of axioms (presumably deriving from the fundamental list), subject to the same constraints as mentioned in the last Ramble.

Peano, player )

Ramble Contents
stoutfellow: My summer look (Summer)
At the heart of the Logicist program was a desire to prove that mathematical statements were true, by resting them on absolutely indisputable bedrock - on logic itself. The second of the major schools to arise, Formalism, took a radically different tack.

Jesting Pilate )

Ramble Contents
stoutfellow: My summer look (Summer)
There are two sardonic answers to the puzzle of the Spanish Barber. One is to posit that the barber does not live in the village; one pictures a chain of villages, each drawing its barber from the next one up the hill. The other is that the barber is a woman, and hence does not require shaving. The two principal responses to Russell's Paradox mirror these, after a fashion.

Type Casting )

Ramble Contents
stoutfellow: (Winter)
The year is 1902. The esteemed philosopher/mathematician Gottlob Frege is about to publish the second volume (of a planned three) of his magnum opus, Grundgesetze der Arithmetik ("Basic Laws of Arithmetic"), broadening and deepening the Logicist program. He has just received a letter from a younger colleague, an Englishman by the name of Bertrand Russell.

Two Doctors )

Ramble Contents
stoutfellow: (Winter)
With the emergence of the new geometries and the new algebras, and with the general increase in abstraction that mathematics underwent in the nineteenth century, there came a need to find new foundations for the subject. If Euclid's certainties were to be dispensed with, where were mathematicians to find secure footing? Three main schools of thought emerged; the first, Logicism, saw in the work of Cantor a new beginning.

Logicism: First Try )

Ramble Contents
stoutfellow: (Winter)
There is no largest cardinality; no matter how large a set is, there is a larger one. The proof of this, which is an adaptation of Cantor diagonalization, is under the cut.

More Power! )

Ramble Contents
stoutfellow: (Winter)
It was Galileo who had observed that there are just as many perfect squares as there are positive whole numbers - that, in other words, a whole (all the positive whole numbers) could be equal to a part (those positive whole numbers which happen to be perfect squares). Given that one of Euclid's "Common Notions" was that the whole is greater than the part, this was disturbing, which probably accounts for the failure of Galileo's successors to investigate further. Georg Cantor dared to do so, with even more disturbing results.

Move Like a Bishop )

Ramble Contents
stoutfellow: (Winter)
At least to begin with, Georg Cantor was just interested in the behavior of Fourier series. His studies, though, were to lead to a thorough restructuring of the foundations of mathematics, and touch off a civil war (not over personal pique, as in the case of Newton/Leibniz, but over fundamental issues concerning the nature of mathematics) that still sputters along to this day.

Derived Sets )

Ramble Contents
stoutfellow: (Winter)
Most of Richard Dedekind's contributions to mathematics were in the area of algebra; one of the fundamental constructs used in algebraic number theory1 is the "Dedekind domain". But he also supplied an alternative construction of the real numbers from the rationals, which will be detailed under the cut.

The Subtlest Cut )

1. As opposed to elementary number theory2 and analytic number theory.
2. Despite its name, this is an independent subfield, with techniques and subtleties all its own.

Ramble Contents
stoutfellow: (Winter)
That the real numbers are not the same as the rational came to the attention of mathematicians as a result of a geometrical problem. In the middle of the nineteenth century, the question arose: can we talk about the real numbers without making any reference to geometry? Can we describe them purely in terms of our familiar rational numbers? The answer proved to be yes, and in two distinctly different ways. This post will discuss one of those ways, devised by Cauchy; the other, due to Richard Dedekind, will be the topic of the next.

Bunching Up )

Ramble Contents
stoutfellow: (Winter)
The job of repairing the shaky foundations of analysis fell to the hands of two mathematicians, Augustin Cauchy and Karl Weierstrass. It was an extended process, and it would take a long time to detail its erratic progress; here, I'll simply discuss in outline the final result.

Come a Little Bit Closer )

Ramble Contents
stoutfellow: (Winter)
Joseph Fourier was more a physicist than a mathematician (though the lines between the disciplines were less clear in his day than in ours); his mathematical papers, even his masterpiece Analytic Theory of Heat, were less than rigorous, and many of his claims proved to be false as stated. Nonetheless, with appropriate qualifications and restrictions, they could be made true, and their consequences have been enormous. Many important modern technologies rest on his work; fields of study ranging from physics to linguistics have drawn upon his insights; and - most importantly for our purposes - he had considerable, if indirect, impact on the foundations of mathematics.

Period Piece )

Ramble Contents
stoutfellow: Joker (Default)
I was going to talk about Joseph Fourier next, but I think I'd better drop back a bit and discuss the importance of power series in analysis first.

Approximations: First Try )

Ramble Contents
stoutfellow: My summer look (Summer)
We'll start examining 19th-century developments in analysis shortly, but I want to make a few observations concerning the importance of quantifiers before doing so.

You can get all of it some of the time... )

Ramble Contents
stoutfellow: My summer look (Summer)
The symbolic logic devised by George Boole and his successors was both powerful and flexible, a major step forward. Still, it was limited; the logic used by Aristotle and extended by the Scholastics of the Middle Ages was capable of feats beyond the reach of Boolean logic. It was Charles Sanders Pierce, again, who found the way forward, unifying the two kinds of logic.

Subatomic Particles )

Ramble Contents

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