Posted on 09/24/2003 11:25:56 PM PDT by betty boop
Why do you do that? When did I suggest mathematics was not real?
Mathematics is a very powerful tool for dealing with those aspects of existence which are countable and measurable, and can even be stretched for use with some things which, in the strictest sense, cannot be counted or measured, such as the ratio of the circumference of a circle to its radius (pi) or the ratio of the length of either leg of an isosceles right triangle to the hypotenuse. I find it strange that those who understand mathematics not only fails to find an absolute value for pi and cannot find a unit of measure that can measure both a leg and hypotenuse of an isosceles right triangle, (and these are just very common limitations of mathematics) believe mathematics is some super-metaphysical power dictating the nature of the world. It's a very useful intellectual tool, but has not other special significance, except to the superstitious who are always in awe of what they do not understand.
My comment about isosceles right triangles being ubiquitous illustrates exactly what I mean. One of the most common geomatic forms in our world cannot be measure by mathematics. I would say mathematics is quite limited, as useful as it is.
Hank
It appears that Hank Kerchief has determined that irrational numbers such as Pythagoras's Constant refute mathematical Platonism in his worldview.
I however find irrational numbers, Gödel's Incompleteness Theorem and Turing machines that do not halt --- all to be part and parcel of the mathematical structure of ”all that there is.” After all, why we think the mathematical structures must be integers, real numbers, true/false decisions, finite, etc.?
Penrose points to the Mandelbrot set as an example of Platonism – and it continues. I agree, but I believe geometry is even more obvious evidence for mathematical Platonism. For example,
The Equivalence Principle as Symmetry
It's not so much that math is a useful tool but that elegant mathematics have been found to correspond extremely closely to the micro and macro behavior of the universe. Math turns out to be powerfully descriptive. Why this should be so is a deep mystery. Early man must have felt he created mathematics and later perhaps that he had discovered aspects of it but there was no reason, then or now, to suspect that it is so deeply entwined with the fundamental nature of the universe, from the Materialist/Objectivist point-of-view. It is nonetheless true that understanding the math has become increasingly important if we care to understand the universe. We aren't yet there.
I know this is how it seems to you and others, and I am not trying to dissuade you or remove from your experience something which is profound to you.
I do not see it that way, however. It would be a mystery to me if mathematics was not useful in describing and providing the shortcut for understanding many aspects of existence. In fact, however, there is more about the material universe that mathematics is either only fairly useful for, or not useful at all.
At one time it was believed every shape in the universe could be reduced to Descartes' analytic geometry. A lot of them can, but a lot more cannot. For those we had to develop new fields such as topology. Then came along fractals and strange attractors which have enabled us to understand some other kinds of shapes and behavior, but, so far, this "chaotic" math is useless in describing any particular shape or behavior. (The problem with fractals is, one can plug in numbers and create all sorts of interesting patterns, but one cannot find a pattern in nature and determine what numbers to plug in to produce it. The other problem is fractals and strange attractors are both "discrete" iterative functions, and even when patterns seem analogue, they are only "connect the dot" type analogue shapes.
Another place that mathematics can only deal with existence as an approximation, at best, is in relationships, two of which have been mentioned before, pi and the ratio of the hypotenuse of an isosceles triangle to either leg. This latter is greatly misunderstood. The pythagoreans were the first to suggest the apparent "mystical" relationship between numbers and existence. It was the discovery of incommensurables, and the limitations of mathematics at its heart that cured them.
The whole significance of incommensurables is, that there are relationships which can certainly be described for which there is no arithmetic means of describing, thus whole worlds of real things mathematics is totally irrelevant to.
The Proof:
"In a right-angled isosceles triangle, the square on the hypotenuse is double of the square on either side. Let us suppose each side an inch long; then how long is the hypotenuse? Let us suppose its length is m/n inches. Then m2/n2 = 2. If m and n have a common factor, divide it out; then either m or n must be odd. Now m2 = 2n2, therefore m2 is even, therefore m is even; therefore n is odd. Suppose m = 2p. Then 4p2 = 2n2, therefore n2 = 2p2 and therefore n is even, contra hyp. Therefore no fraction m/n will measure the hypotenuse. This proof is substantially that in Euclid, Book 10." (Bertand Russell, A History of Western Philosophy) In other words, there is no even or odd number that can be the measure of the hypotenuse relative to either leg, therefore, there is no such number at all, but there is certainly such a relationship which can be both defined and understood, but not with the help of mathematics.
Hank
Not how it seems, Hank, how it is. That YOU cannot see it is, to me, the issue.
Oh,I'm sure that's not true. What I see or don't see is of no importance at all to you, or should not be. It is what you see that should matter to you.
The bottom line is this. There are two abstract views of knowledge regarding what is significant and what is not. My view may very simply be stated:
What cannot be known cannot matter.
Now notice, I did not say what is not known, but what cannot be known.
Most people really believe the opposite. It is always interesting to have people who disagree with my view attempt to provide examples of what cannot be known which are, nevertheless, important. Of course, if something cannot be known, it certainly cannot be provided as an example, because it would not be known.
Hank
So mathematics is a language with a built-in logic. But what is so striking about this language is that it seems to describe how the world works—not just sometimes, not just approximately, but invariably and with unfailing accuracy. All the fundamental sciences—physics, chemistry, and astronomy—are mathematical sciences. No phenomenon has ever been discovered in these subjects for which a mathematical description is not only possible but also beautifully appropriate. Yet one could still fail to be impressed. After the fact, perhaps, we can force any hand into some glove, and maybe we have chosen to pick the mathematical glove because it is the only one available. It is striking, however, that physicists so often find that some esoteric mathematical structure, invented by mathematicians in the dim and distant past only for the sake of its elegance and curiosity value, is precisely what is required to make sense of new observations of the world. In fact, confidence in mathematics has grown to such an extent that one now expects (and finds) interesting mathematical structures to be deployed in nature. Scientists look no further when they have found a mathematical explanation.
There are many striking examples of the unexpected and curious effectiveness of mathematics. In 1914, when Einstein was struggling to formulate a new description of gravity to supersede that of Newton, he wished to endow the universe with curved space and time, and to codify the laws of nature in a manner that would apply for any observers no matter what their state of motion. His old student friend, the mathematician Marcel Grossman, introduced him to a little-known branch of nineteenth-century mathematics, called tensor calculus, that was tailor-made for his purposes. Upon adopting this mathematical language, how Einstein would describe laws of nature became clear and (if one is that clever) obvious.
In modern times, particle physicists have discovered that symmetry dictates the way elementary particles behave. Particular collections of related particles can behave in any way they choose so long as a particular abstract pattern is preserved. The laws of nature are superficially the catalog of habitual things that occur in the world while yet preserving these patterns. With every such catalog of changes one can always find an unchanging pattern, though the pattern is often subtle and rather abstract.
Intelligibility of the Universe - Gregory Chaitin
Abstract: We discuss views about whether the universe can be rationally comprehended, starting with Plato, then Leibniz, and then the views of some distinguished scientists of the previous century. Based on this, we defend the thesis that comprehension is compression, i.e., explaining many facts using few theoretical assumptions, and that a theory may be viewed as a computer program for calculating observations. This provides motivation for defining the complexity of something to be the size of the simplest theory for it, in other words, the size of the smallest program for calculating it. This is the central idea of algorithmic information theory (AIT), a field of theoretical computer science. Using the mathematical concept of program-size complexity, we exhibit irreducible mathematical facts, mathematical facts that cannot be demonstrated using any mathematical theory simpler than they are. It follows that the world of mathematical ideas has infinite complexity and is therefore not fully comprehensible, at least not in a static fashion. Whether the physical world has finite or infinite complexity remains to be seen. Current science believes that the world contains randomness, and is therefore also infinitely complex, but a deterministic universe that simulates randomness via pseudo-randomness is also a possibility, at least according to recent highly speculative work of S. Wolfram. [Written for a meeting of the German Philosophical Society, Bonn, September 2002.]
Hank, with regard to your assertion of Pythagoras's Constant, I would like to add that "it is not known if Pythagoras's constant is normal to any base": Normal Numbers
OK Hank, how can one determine what cannot be known? Seems to me we've done pretty well expanding the sphere of the known in the face of the naysayers over recent centuries. As to philosophy, there is no basis for the statement. What cannot be known may matter greatly, as a function of course of what "to matter" means (no, I am not a Bill Clinton wannabe -- where are the lawyers when we need them?).
OK Hank, how can one determine what cannot be known? Seems to me we've done pretty well expanding the sphere of the known in the face of the naysayers over recent centuries. As to philosophy, there is no basis for the statement. What cannot be known may matter greatly, as a function of course of what "to matter" means (no, I am not a Bill Clinton wannabe -- where are the lawyers when we need them?).
You have restored my faith in philosophers. The "principle" is sort of a trap for those who want to make some unknowable the ultimate "principle" or something. Like those who make God the ultimate reason for everything, but then declare "He cannot be known." It is neither in opposition to God, or any other principle, only the idea that "what cannot be known," can be part of any answer.
You are absolutely right, it has nothing to do with philosophy, at least any correct philosophy, and you are also correct, that men have leaned much, and will continue to do so.
I wasn't trying to trick you, by the way, simply interested in your response, which pleased me much.
Thanks! Hank
For example, we cannot know the future, at least not until it has arrived.
We cannot know what another's conscious experience is like. Even if you describe yours to me, I must interpret it in terms of my own conscious experience, which might be the same as yours or not, but there is no way to know.
I would even adventure to say, there are probably more things we cannot know than there are that we can, simply because we are finite, and there infinite things to know. (This, of course, is very abstract, because it does not specify any particular class of things we cannot know.)
Maybe you can come up with some other things we cannot know. I think it would be an iteresting experiment, and possibly even useful, eliminating from the field of enquirey those things that cannot be known so we can concentrate on those that can be known.
Hank
From a purely scientific reality we are "Ugly bags of mostly water".
LOL!!! Please do feel free to speak for yourself, PigRigger!
p.s.: What scientific "reality?" Sounds more like scientism speaking. In other words, just another cheap dogma -- I've noticed that nouns ending with the suffix, "ism," usually denote some kind of dogma and, therefore, some degree of irrationality in their adherents. Just call it a "rule of thumb" that I find useful.
How do you know what cannot be known? Do you mean to suggest that what you feel you "cannot know" is ultimately, truly unknowable on principle?
For instance, Hank says the future cannot be known. But math and physical laws work against that view; for instance, as soon as I press "post" I shall add 2 plus 2 in decimal and I know the answer will be 4 in decimal.
Further, I perceive ”all that there is” to include extra dimensions, one of which is time, and therefore time is a plane and not a line and thus, not only is the future, past and present known in that dimension but it may also be knowable in 4D.
Likewise, by the grace of God, I know Him and perceive many things which are not knowable to one who does not have the “eyes to see” or the “ears to hear.” But others who share the same Holy Spirit are of the same mind; we perceive spiritual matters as One being.
Which things also we speak, not in the words which man's wisdom teacheth, but which the Holy Ghost teacheth; comparing spiritual things with spiritual. But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know [them], because they are spiritually discerned.
But he that is spiritual judgeth all things, yet he himself is judged of no man. For who hath known the mind of the Lord, that he may instruct him? But we have the mind of Christ. - I Corinthians 2:11-16
So what would be the point of trying to define "what cannot be known"?
At the end, when Chaitin compares notes to Wolfram and pi - it is good to keep in mind that Chaitin is the one who discovered omega... (wink)
Yes, we can know the future contingently, on the basis of, if this then thus'n'so, but, as soon as you press post you may drop dead (heaven forbid) or the power on your calclator may go out and you will not do what you intend. You just don't know.
...But we have the mind of Christ ...
Yes, but which. The mind of Christ as the risen ascended Redeemer, or the mind of Christ within the limits of human capacity, just as His was while in this world, for though He was given the Spirit without measure (John 3:34) yet He could not tell if there were figs on a tree without looking (Mark 11:13). So maybe one of the things we cannot know is whether or not there is fruit on a tree, at least without looking. (It is similar to the cat in the cupboard, if you followed that earlier).
So what would be the point of trying to define "what cannot be known"?
Possibly none beyond an interesting exercise. Surely there would be some things we all would agree cannot be known, and those we disagreed on might provide insights into why we disagree, and what the differences in our understanding of knowledge are really based on.
You've already provided some interesting insights.
Hank
Of course we cannot know what cannot be known, but we can know some kinds of things which cannot be known. I suggested a couple in post #113. We cannot know things which are just not available to us to know. We cannot know factually any detail of the universe beyond a certain distance, because the light from those parts of the universe has not reached us yet. We can conjecture and make intelligent guesses what it might be like in general, but cannot know the detail.
It has nothing to do with feeling.
Hank
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