Registered: Aug 2003
Posts: 712

Bram Boroson talked about physics ideas online at NKS 2004. He included a quote from Einstein on the subject of possible discrete models of physics. Several people have said it was an interesting quote, so I am posting it here.

"To be sure, it has been pointed out that the introduction of a space-time continuum may be considered as contrary to nature in view of the molecular structure of everything which happens on a small scale. It is maintained that perhaps the success of the Heisenberg method points to a purely algebraical method of description of nature, that is to the elimination of continuous functions from physics. Then, however, we must also give up, by principle, the space-time continuum. It is not unimaginable that human ingenuity will some day find methods which will make it possible to proceed along such a path. At the present time, however, such a program looks like an attempt to breathe in empty space." - Albert Einstein, Out of My Later Years, page 92

I hope this is interesting.

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Registered: Feb 2004
Posts: 551

Jason,

Being by nature a combinatorial thinker,
I am tempted to just say "Bah! Humbug!"
or "Discreteness Forever!" and let it go
at that, but one of the issues behind the
issue is the difference between a reality
and a model.

The fact is that our models are always
approximations to our realities, and the
kicker is this: Sometimes a discrete model
is a good approach to a continuous reality
and sometimes a continuous model is a good
approach to a discrete reality.

Just two examples of the later case:

In statistics, we may know for certain
that our population is discrete and finite,
but when N grows beyond 30 or so, a sensible
person will put away the binomial distribution
tables and use the normal distribution instead.

For similar reasons in probabilistic number theory,
where the population is discrete but infinite,
a lot of information is most easily gained
by passing to the limit in the model.

So maybe Uncle Albert had something after all ...

Jon Awbrey

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Registered: Jul 2004
Posts: 2

Hope This quote from Einstein would also be useful

"So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality"

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Registered: Aug 2003
Posts: 712

The problem is that Einstein is there just retailing an idea from Kant, itself in reaction to skeptical arguments from Hume. "Certainty" is simply not a useful concept in these contexts. Its meaning is largely a matter of philosophical commitments if not semantic hairsplitting, not a matter of theory.

If you define certainty such that there isn't any, you put brackets around all your statements but they remain otherwise unchanged. If you define certainty such that there isn't any in physical theory but can be some in mathematics, then you just repeat a distinction between physical theory and mathematics. Which can be misleading if e.g. it suggests that everything is clear, simple, and certain as long as it remains on the terrain of mathematics - a common enough notion that just happens to be completely false. (Plenty of math is hard).

At best the distinction ("certain" or "uncertain" I mean) is unhelpful. Because the real distinction is between cases simple enough to be analyzed completely, whether they are physical or mathematical or logical, and cases that just aren't simple enough to be so analyzed, which again can arise in any of those areas. At worst, it is an attempt to legislate as the meaning of a term, a single philosophical view of what causes limits to knowledge (the idea that the only hard part is getting outside our heads), that as it happens begs most of the interesting questions.

Terms should clarify alternate views of what issues matter, not try to decide between them. Personally I avoid "certainty" as a subject. But I reserve its (to me largely vacuous) use to the Humean skeptic context, where it is one of those "all is x" concepts that does not distinguish between one thing and another, but instead lumps every real or ideal referrent together. What is certain, not a darn thing, in this sense of the term. Making it a less than riveting distinction, since it fails to distinguish.

Kant, in constrast, wanted to allow entirely analytic propositions to be certain, thus putting logic and math on one side, and every statement about the phenomenal world on the other. Einstein is essentially doing the same in your quote. This is a different possible meaning to assign to the term "certain". It does make a real distinction, in that usage. But that distinction assumes logic and math are easier than they are, or that nothing slippery will happen in the purely analytic realm. Which is not one possible meaning to assign terms, but just plain false.

If that seems like an excessive claim, simply particularize it and it will seem like mere common sense. The Goldbach conjecture (binary or strong version - any even number greater than 2 is the sum of 2 primes) is an analytic proposition in arithmetic. So, is it supposed to be certain? Nobody has found a counterexample, even with the aid of modern computers. (There is none less than 6x10^16). And no mathematician has proved it in 160 years. Prizes of $1 million have been offered for a proof, but not collected. No sensible usage would call this "certainty".

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Registered: Jan 2009
Posts: 6

My brain isn't up to complicated philosphical dueling right now, but... I have never liked Einstein's way (or Poincare's) way of saying that physical "theories" are only free creations of the human mind. I agree that room for questions ought always to be made, and that our theories often turn out not to match reality exactly, but the psychic impulse for scientific discovery is that ultimate order exists. Einstein himself made the same in objecting to quantum physics, "I cannot believe that God plays dice with the universe." I object to that statement as well, because Einstein was not serious about any kind of belief in God, and the mathematics of probability are not statements about uncertainty, they are only statements about the number of ways that particles can be arranged on a level where the exact configuration cannot be precisely known because of the complexity, not because of an ultimately random universe.

Although Einstein's comments seem meant to be taken lightly, they appear to me to be attempts to justify inconsistencies in his own theorys and at the same time a way to take aim at those he considers potentially contrary to his own. I used to be an admirer of Einstein, but I found myself unable to accept certain key inconsistencies in his logic, as quite often, right after establishing the existence of certain physical laws, he came about and turned those upside down. All of this was done so as to contrive multiple "frames of reference" so that different observors could disagree on simultaneity while agreeing on the constancy of light speed. Others have noted that the lack of agreement for simultaneity depends solely on Einstein's definition, while others maintain that there are other ways to account for the constancy (or lack of constancy) of light speed.

The reason for this comment is to draw attention to these odd comments as well as the increasing numbers of those who take issue with Einstein for similar reasons. I would feel particularly ill suited to make critical comments being not (at least for the moment) able to follow the philosophical fine points of this discussion. Still, for a short while I was focusing on these oddities and seemingly fatal contradictons in Einstein's concepts, when it suddenly occurred to me why he was wrong. Actually, I had always been convinced of that, but held back because I couldn't find any correct alternatives. That changed for me a bit over a year ago, however, and I felt confident, and still do, that I had found the "Unified Field," at least as Einstein thought of it.

What I am referring to is a proper way of interpreting the Michelson-Morley experiment without having to invoke the Lorentz transformations as coordinate frames or invoke the concept of light speed in the same manner. I'm also talking about a construction that eliminates the contradictions that people frequently point out, and one that makes it possible to account for known discrepancies between observation and mathematical prediction... so, while not a physicist, but a reasonably good mathematician, this is the claim that I make. If anyone wishes to inspect the claim or make comment, feel free to do so. The link to that paper is below, but I want to say that the randomized block theorem is my real baby and I see no interest on that subject here on NKS, so statisticians are welcom. Otherwise, click on the Unified button...

http://foossolvesunified.com
--- thank you very much! Alan



=====================================

Originally posted by Jason Cawley
The problem is that Einstein is there just retailing an idea from Kant, itself in reaction to skeptical arguments from Hume. "Certainty" is simply not a useful concept in these contexts. Its meaning is largely a matter of philosophical commitments if not semantic hairsplitting, not a matter of theory.

If you define certainty such that there isn't any, you put brackets around all your statements but they remain otherwise unchanged. If you define certainty such that there isn't any in physical theory but can be some in mathematics, then you just repeat a distinction between physical theory and mathematics. Which can be misleading if e.g. it suggests that everything is clear, simple, and certain as long as it remains on the terrain of mathematics - a common enough notion that just happens to be completely false. (Plenty of math is hard).

At best the distinction ("certain" or "uncertain" I mean) is unhelpful. Because the real distinction is between cases simple enough to be analyzed completely, whether they are physical or mathematical or logical, and cases that just aren't simple enough to be so analyzed, which again can arise in any of those areas. At worst, it is an attempt to legislate as the meaning of a term, a single philosophical view of what causes limits to knowledge (the idea that the only hard part is getting outside our heads), that as it happens begs most of the interesting questions.

Terms should clarify alternate views of what issues matter, not try to decide between them. Personally I avoid "certainty" as a subject. But I reserve its (to me largely vacuous) use to the Humean skeptic context, where it is one of those "all is x" concepts that does not distinguish between one thing and another, but instead lumps every real or ideal referrent together. What is certain, not a darn thing, in this sense of the term. Making it a less than riveting distinction, since it fails to distinguish.

Kant, in constrast, wanted to allow entirely analytic propositions to be certain, thus putting logic and math on one side, and every statement about the phenomenal world on the other. Einstein is essentially doing the same in your quote. This is a different possible meaning to assign to the term "certain". It does make a real distinction, in that usage. But that distinction assumes logic and math are easier than they are, or that nothing slippery will happen in the purely analytic realm. Which is not one possible meaning to assign terms, but just plain false.

If that seems like an excessive claim, simply particularize it and it will seem like mere common sense. The Goldbach conjecture (binary or strong version - any even number greater than 2 is the sum of 2 primes) is an analytic proposition in arithmetic. So, is it supposed to be certain? Nobody has found a counterexample, even with the aid of modern computers. (There is none less than 6x10^16). And no mathematician has proved it in 160 years. Prizes of $1 million have been offered for a proof, but not collected. No sensible usage would call this "certainty".

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