Tuesday, June 26, 2007

On Hilbert's sixth problem

The world of null-H post is the next post down.

There is no consensus on whether Hilbert's sixth problem: Can physics be axiomatized? has been answered.

From Wikipedia, we have this statement attributed to Hilbert:

6. Mathematical treatment of the axioms of physics. The investigations of the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.

Hilbert proposed his problems near the dawn of the Planck revolution, while the debate was raging about statistical methods and entropy, and the atomic hypothesis. It would be another five years before Einstein conclusively proved the existence of atoms.

It would be another year before Russell discovered the set of all sets paradox, which is similar to Cantor's power set paradox. Though Cantor uncovered this paradox, or perhaps theorem, in the late 1890s, I am uncertain how cognizant of it Hilbert was.

Interestingly, by the 1920s, Zermelo, Fraenkel and Skolem had axiomatized set theory, specifically forbidding that a set could be an element of itself and hence getting rid of the annoying self-referencing issues that so challenged Russell and Whitehead. But, in the early 1930s, along came Goedel and proved that ZF set theory was either inconsistent or incomplete. His proof actually used Russell's Principia Mathematica as a basis, but generalizes to apply to all but very limited mathematical systems of deduction. Since mathematical systems can be defined in terms of ZF, it follows that ZF must contain some theorems that cannot be tracked back to axioms. So the attempt to axiomatize ZF didn't completely succeed.

In turn, it would seem that Goedel, who began his career as a physicist, had knocked the wind out of Problem 6. Of course, many physicists have not accepted this point, arguing that Goedel's incompleteness theorem applies to only one essentially trivial matter.

In a previous essay, I have discussed the impossibility of modeling the universe as a Turing machine. If that argument is correct, then it would seem that Hilbert's sixth problem has been answered. But I propose here to skip the Turing machine concept and use another idea.

Conceptually, if a number is computable, a Turing machine can compute it. Then again Church's lamda calculus, a recursive method, also allegedly could compute any computable. So are the general Turing machine and the lamda calculus equivalent? Church's thesis conjectures that they are, implying that it is unknown whether either misses some computables (rationals or rational approximations to irrationals).

But Boolean algebra is the real-world venue used by computer scientists. If an output can't be computed with a Boolean system, no one will bother with it. So it seems appropriate to define an algorithm as anything that can be modeled by an mxn truth table and its corresponding Boolean statement.

The truth table has a Boolean statement where each element is above the relevant column. So a sequence of truth tables can be redrawn as a single truth table under a statement combined from the sub-statements. If a sequence of truth tables branches into parallel sequences, the parallel sequences can be placed consecutively and recombined with an appropriate connective.

One may ask about more than one simultaneous output value. We regard this as a single output set with n output elements.

So then, if something is computable, we expect that there is some finite mxn truth table and corresponding Boolean statement. Now we already know that Goedel has proved that, for any sufficiently rich system, there is a Boolean statement that is true, but NOT provably so. That is, the statement is constructible using lawful combinations of Boolean symbols, but the statement cannot be derived from axioms without extension of the axioms, which in turn implies another statement that cannot be derived from the extended axioms, ad infinitum.

Hence, not every truth table, and not every algorithm, can be reduced to axioms. That is, there must always be an algorithm or truth table that shows that a "scientific" system of deduction is always either inconsistent or incomplete.

Now suppose we ignore that point and assume that human minds are able to model the universe as an algorithm, perhaps as some mathematico-logical theory; i.e., a group of "cause-effect" logic gates, or specifically, as some mxn truth table. Obviously, we have to account for quantum uncertainty. Yet, suppose we can do that and also suppose that the truth table need only work with rational numbers, perhaps on grounds that continuous phenomena are a convenient fiction and that the universe operates in quantum spurts.

Yet there is another proof of incompleteness. The algorithm, or its associated truth table, is an output value of the universe -- though some might argue that the algorithm is a Platonic idea that one discovers rather than constructs. Still, once scientists arrive at this table, we must agree that the laws of mechanics supposedly were at work so that the thoughts and actions of these scientists were part of a massively complex system of logic gate equivalents.

So then the n-character, grammatically correct Boolean statement for the universe must have itself as an output value. Now, we can regard this statement as a unique number by simply assigning integer values to each element of the set of Boolean symbols. The integers then follow a specific order, yielding a corresponding integer.
(The number of symbols n may be regarded as corresponding to some finite time interval.)

Now then, supposing the cosmos is a machine governed by the cosmic program, the cosmic number should be computable by this machine (again the scientists involved acting as relays, logic gates and so forth). However, the machine needs to be instructed to compute this number. So the machine must compute the basis of the "choice." So it must have a program to compute the program that selects which Boolean statement to use, which in turn implies another such program, ad infinitum.

In fact, there are two related issues here: the Boolean algebra used to represent the cosmic physical system requires a set of axioms, such as Hutchinson's postulates, in order to be of service. But how does the program decide which axioms it needs for itself? Similarly, the specific Boolean statement requires its own set of axioms. Again, how does the program decide on the proper axioms?

So then, the cosmos cannot be fully modeled according to normal scientific logic -- though one can use such logic to find intersections of sets of "events." Then one is left to wonder whether a different system of representation might also be valid, though the systems might not be fully equivalent.

At any rate, the verdict is clear: what is normally regarded as the discipline of physics cannot be axiomatized without resort to infinite regression.

So, we now face the possibility that two scientific systems of representation may each be correct and yet not equivalent.

To illustrate this idea, consider the base 10 and base 2 number systems. There are some non-integer rationals in base 10 that cannot be expressed in base 2, although approximation can be made as close as we like. These two systems of representation of rationals are not equivalent.

(Cantor's algorithm to compute all the rationals uses the base 10 system. However, he did not show that all base n rationals appear in the base 10 system.)


At 6:41 PM , Blogger AL said...

Hilbert's Sixth has been solved.


At 1:42 PM , Anonymous Anonymous said...

A) Your assertion that ZF must contain some theorems which annot be proved is rather sloppy. A theorem is a proposition which can be proved. You meant to say that ZF contains some propositions which cannot be proved, and whose negations cannot be proved either. This is incompleteness.

B) Goedel's Incompleteness Theorem has nothing to do with whether Physics can be axiomatised. It might have something to do with whether Theoretical Physics can ever be complete. But it might not. After all, Theoretical Physics does not have to include all of ZF. Set theory might go way beyond physical reality, so that Theoretical Physics would not actually contain all of mathematics, and thus Goedel's theorem would not apply. In fact, Goedel also proved a completeness theorem: First order Logic is complete. Now, why would a physicist want second-order quantifiers? So, who knows.

C) Forget Boolean algorithms and Turing Machines. No algorithm can be exactly modelled by a physical machine: there is always some noise. E.g., no square wave, and hence no string of bits, can be completely reliably produced in the real world, and if it could be produced, the information in it, the «signal», still could not be reliably extracted in an error-free fashion by any finite physical apparatus.


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