On 3/14/2013 5:43 AM, david petry wrote: > On Thursday, March 14, 2013 3:17:06 AM UTC-7, fom wrote: > >> For example, Wittgenstein understood perfectly >> well how to apply Cantor's argument and he >> certainly is not thought of as believing in >> a completed infinity. > > > Here's an "acceptable" use of the diagonal argument: Given a provably well-defined list of provably well-defined real numbers (so that every digit of every number on the list can provably be computed), the diagonal argument gives us a new provably well-defined real number not on the list. > > Notice that that argument doesn't require the use of an actual infinite. >
Right. I have a preprint of a text on enumerable sets written long ago by Robert Soare. In the opening pages he discusses the diagonalizability of total recursive functions. So, there is an "acceptable" diagonal argument for mathematics restricted to recursive function theory. Soare goes on to observe that the total functions do not span every computable function so that partial recursive functions are introduced.
> > >> However, he also did not attack the mathematicians >> who conducted investigations along those lines. > > He most certainly did mock them. >
Perhaps. I have a small book transcribing certain lectures he gave to certain notable mathematicians. It was civil. He gave wonderful explanations of how natural language structure accommodates the typical beliefs of mathematicians as they work with abstract objects.
The problem is that "language game" theories do not help to explain why a bridge of building does not crash to the ground. Early pyramids do not have the majesty of those built later at Gizeh. Once one begins the process of using mathematics for real-world situations, one begins referring to mathematical terms with the same linguistic forms as one would with material objects like chairs and tables.
Next, you have a 7 year old child asking "Why?"
Because physics has seemingly given up on explaining "the observable universe" in favor of mathematical abstractions, non-material physicalism is becoming the only reasonable choice.
As for mathematics, I have no reason to support its uses to discredit anyone's metaphysical beliefs. It is perfectly plausible to treat mathematics along the lines of Lesniewskian nominalism and regard its essential value in the interderivability of its facts. That is, because mathematics can be understood in the sense of an Aristotelian demonstrative science, there is no reason to confuse its facticity with actuality.
More precisely, mathematical facticity relative to the epistemic justification of a derivational system is more important than mathematical actuality relative to dialectical argument. Thanks to modern advances beginning in the nineteenth century, it is possible to divorce "essence" from "substance" in the Aristotelian framework.
Modern logicians, however, try to distance themselves from epistemology. Therefore, one has only the Aristotelian distinction to guide this interpretation.
> > FWIW, I was motivated to start this thread after reading an article you wrote about problems you had in your encounters with mathematicians who didn't accept your ideas. I'd like to know a little more about that, although I admit you really have no obligation to tell us more if you don't want to. > > The mathematics community has simply lost touch with reality. > > >
I have no real issues along those lines except that sometimes the personal frustration becomes somewhat emotional. Professional mathematicians are busy people. Most who might be interested in "pure mathematics" have responsibilities that must be met to justify the salaries they receive. On the one hand, they are expected to teach. To the extent that they receive additional time for their own interests, they are expected to publish. That latter expectation is demanded from the same people who think quarterly profits are important to the security of the endowments with which they are entrusted.
Because I had been unable to fulfill my own academic goals, I can only hope that perhaps someone would take an interest in my work. Given the reality of the world, I probably have a better chance of winning a lottery (which, of course, is less likely than being hit by lightning walking in a grassy field during a thunderstorm).
There is even more to consider. If common accounts of Greek mathematics as the first mathematics that can be considered "abstract" are reliable, then mathematics is a 2500 year old subject. Any given mathematician knows only a small part of what that entails and even less of how mathematics is being practically applied. Almost anyone who investigates a particular aspect of mathematics that will require investigation into the historical record will be surprised at what they find.
In "A Mathematical History of the Golden Number," Roger Herz-Fischler writes:
"At first it seemed as if this mathematical history would be fairly short and straightforward. This opinion was based on preliminary reading not only of parts of the Elements [Euclid], but also some of the standard histories of Greek mathematics. However, two things soon became clear: the early Greek aspect was not as clear-cut as it was often made out to be and the historical aspects that needed to be considered neither started nor ended with the early Greeks."
But, then if certain more informed accounts are to be believed, I have mislead you. In "Men of Mathematics" by E. T. Bell we find the remark,
"[...] Finally we arrive at the first great age of mathematics, about 2000 B.C, in the Euphrates Valley.
"The descendants of the Sumerians in Babylon appear to have been the first "moderns" in mathematics; certainly their attack on algebraic equations is more in the spirit of the algebra we know than anything done by the Greeks in their Golden Age. More important than the technical algebra of those ancient Babylonians is their recognition -- as shown by their work -- of the necessity for *proof* in mathematics. Until recently it had been supposed that the Greeks were the first to recognize that proof is demanded for mathematical propositions. This is one of the most important steps ever taken by human beings. Unfortunately it was taken so long ago that it led nowhere in particular so far as our own civilization is concerned -- unless the Greeks followed consciously, which they may well have done. They were not particularly generous to their predecessors."
How could anyone expect a handful of men to be expert on the full array of mathematical knowledge? And, how could anyone expect men and women to be different from the way men and women have to be in order to live from day to day and succeed in their personal pursuits? I would be irrational if I failed to recognize just how small the small indignations I have experienced really are.
Whatever difficulties I may have had through the years, I was educated in a respectable program. I respect the people who study mathematics. I respect mathematics as an academic discipline. And, although one has to view one's own interest in particular ways, I try to respect other mathematician's interests by trying to understand what they find so interesting in what they study.
In view of my general isolation from actively practicing mathematicians, however, all of that effort is simply personal effort to learn through reading.
You may find some justifications for your beliefs in comments you might read that I have written. And, I might sometimes be sharp with my remarks to others. But, you will not find me sympathetic to your interpretations in this regard.