Tom Caylor wrote:
> 1Z wrote:
>> Tom Caylor wrote:
>>> 1Z wrote:
>>>> Tom Caylor wrote:
>>>>> David and 1Z:
>>>>>
>>>>> How is exploring the Mandelbrot set through computation any different
>>>>> than exploring subatomic particles through computation (needed to
>>>>> successively approach the accuracies needed for the collisions in the
>>>>> linear accelerator)? Is not the only difference that in one case we
>>>>> have a priori labeled the object of study 'matter' and in the other
>>>>> case a 'set of numbers'? Granted, in the matter case we need more
>>>>> energy to explore, but couldn't this be simply from the sheer quantity
>>>>> of "number histories" we are dealing with compared to the Mandelbrot
>>>>> set?
>>>>>
>>>>> Tom
>>>>
>>>>
>>>> A number of recent developments in mathematics, such as the increased
>>>> use of computers to assist proof, and doubts about the correct choice
>>>> of basic axioms, have given rise to a view called quasi-empiricism.
>>>> This challenges the traditional idea of mathematical truth as eternal
>>>> and discoverable apriori.
>>> In either case, with math and matter, our belief is that there is an
>>> eternal truth to be discovered, i.e. a truth that is independent of the
>>> observer.
>> "Eternal" doesn't mean "independent of the observer".
>> Empirically-detectable facts are often fleeting.
>>
>
> We're getting into the typical bifurcation of interpretation of terms.
> When you used the term "eternal" to describe math truth, I assumed you
> were talking about something that is independent of time.
>
>>>> According to quasi-empiricists the use of a
>>>> computer to perform a proof is a form of experiment. But it remains the
>>>> case that any mathematical problem that can in principle be solved by
>>>> shutting you eye and thinking. Computers are used because mathematians
>>>> do not have infinite mental resources; they are an aid.
>>> In either case, an experiment is a procedure that is followed which
>>> outputs information about the truth we are trying to discover. Math
>>> problems that we can solve by shutting our eyes are solvable that way
>>> because they are simple enough. As you point out, there are math
>>> problems that are too complex to solve by shutting our eyes. In fact
>>> there are math problems which are unsolvable. I think Bruno
>>> hypothesizes that the frontier of solvability/unsolvability in
>>> math/logic is complex enough to cover all there is to know about
>>> physics. Therefore, what role is left for matter?
>> Physical truth is a tiny subset of mathematical truth.
>>
>
> This agrees with what I am saying.
>
>>>> Contrast this
>>>> with traditonal sciences like chemistry or biology, where real-world
>>>> objects have to be studied, and would still have to be studied by
>>>> super-scientitists with an IQ of a million. In genuinely emprical
>>>> sciences, experimentation and observation are used to gain information.
>>>> In mathematics the information of the solution to a problem is always
>>>> latent in the starting-point, the basic axioms and the formulation of
>>>> the problem. The process of thinking through a problem simply makes
>>>> this latent information explicit. (I say simply, but of ocurse it is
>>>> often very non-trivial).
>>> The belief about matter is that there are basic properties of matter
>>> which are the starting point for all of physics, and that all of the
>>> outcomes of the sciences are latent in this starting point, just as in
>>> mathematics.
>> You can't deduce the state of the universe at
>> time T in any detailed way from the properties of matter,
>
> This is a subject of debate.
>
>> you have to
>> get
>> out your telescope and look.
>>
>
> A telescope could be a way of looking at the state of the computation
> of the universe. This doesn't preclude being able to in theory compute
> the universe (in 3rd pov).
>
>>>> The use of a computer externalises this
>>>> process. The computer may be outside the mathematician's head but all
>>>> the information that comes out of it is information that went into it.
>>>> Mathematics is in that sense still apriori.
>>>> Having said that, the quasi-empricist still has some points about the
>>>> modern style of mathematics. Axioms look less like eternal truths and
>>>> mroe like hypotheses which are used for a while but may eventualy be
>>>> discarded if they prove problematical, like the role of scientific
>>>> hypotheses in Popper's philosophy.
>>>>
>>>> Thus mathematics has some of the look and feel of empirical science
>>>> without being empricial in the most essential sense -- that of needing
>>>> an input of inormation from outside the head."Quasi" indeed!
>>> I'd say that the common belief of mathematicians is that axioms are
>>> just a (temporary) framework with which to think about the invariant
>>> truths.
>> The "truths" are not invariant with regard to choice
>> of axioms. Consider Euclid's fifth postulate.
>>
>
> Euclid's fifth postulate is an axiom.
>
>>> And one of the most important (unspoken) axioms is the
>>> convenient "myth" that I don't need any input from outside my head, so
>>> that I can have "total" control of what's going on in my head, an
>>> essential element for believing the outcome of my thinking. However,
>>> the fact is that a mathematician indeed would not be able to discover
>>> anything about math without external input at some point. This is the
>>> process of learning to think.
>> You need to learn axioms and rules of inference. Everything else
>> is implicit in them.
>>
>
> Discovery is not simply a matter of seeing where a particular set of
> axioms and rules of inference leads. It's only when you see the truth
> from different perspectives, i.e. different sets of axioms and rules of
> inference, that you can start putting together a picture that gets
> closer and closer to reality.
>
> Tom
It's even more than seeing where axioms and rules of inference lead. Given some axioms and rules of inference the only truths you can reach are those of the form "It is true that axioms => theorems".
Brent Meeker
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Received on Tue Oct 24 2006 - 19:21:23 PDT