Friday, October 13, 2017

Reductionism provably fails in mathematics. Can it succeed in science?

From John Lennox's God's  Undertaker, 52-53. 

The great mathematician David Hilbert, spurred on by the singular
achievements of mathematical compression, thought that the reductionist
programme of mathematics could be carried out to such an extent that in
the end all of mathematics could be compressed into a collection of formal
statements in a finite set of symbols together with a finite set of axioms and
rules of inference. It was a seductive thought with the ultimate in ‘bottom-up’
explanation as the glittering prize. Mathematics, if Hilbert’s Programme
were to succeed, would henceforth be reduced to a set of written marks
that could be manipulated according to prescribed rules without any
attention being paid to the applications that would give ‘significance’ to
those marks. In particular, the truth or falsity of any given string of symbols
would be decided by some general algorithmic process. The hunt was
on to solve the so-called Entscheidungsproblem by finding that general
decision procedure.

Experience suggested to Hilbert and others that the Entscheidungsproblem
would be solved positively. But their intuition proved wrong. In 1931
the Austrian mathematician Kurt Godel published a paper entitled ‘On
Formally Undecidable Propositions of Principia Mathematica and Related
Systems’. His paper, though only twenty-five pages long, caused the
mathematical equivalent of an earthquake whose reverberations are still
palpable. For Godel had actually proved that Hilbert’s Programme was
doomed in that it was unrealizable. In a piece of mathematics that stands
as an intellectual tour-de-force of the first magnitude, Godel demonstrated
that the arithmetic with which we are all familiar is incomplete: that is,
in any system that has a finite set of axioms and rules of inference and
which is large enough to contain ordinary arithmetic, there are always true
statements of the system that cannot be proved on the basis of that set of
axioms and those rules of inference. This result is known as Godel’s First
Incompleteness Theorem.

Now Hilbert’s Programme also aimed to prove the essential consistency
of his formulation of mathematics as a formal system. Godel, in his
Second Incompleteness Theorem, shattered that hope as well. He proved
that one of the statements that cannot be proved in a sufficiently strong
formal system is the consistency of the system itself. In other words, if
arithmetic is consistent then that fact is one of the things that cannot be
proved in the system. It is something that we can only believe on the basis
of the evidence, or by appeal to higher axioms. This has been succinctly
summarized by saying that if a religion is something whose foundations
are based on faith, then mathematics is the only religion that can prove it
is a religion!

In informal terms, as the British-born American physicist and
mathematician Freeman Dyson puts it, ‘Godel proved that in mathematics
the whole is always greater than the sum of the parts’.10 Thus there is a limit
to reductionism. Therefore, Peter Atkins’ statement, cited earlier, that ‘the
only grounds for supposing that reductionism will fail are pessimism in
the minds of the scientists and fear in the minds of the religious’ is simply

incorrect.

15 comments:

Stardusty Psyche said...

" In other words, if arithmetic is consistent then that fact is one of the things that cannot be proved in the system. It is something that we can only believe on the basis of the evidence, or by appeal to higher axioms."
--Right. The principles of logic are not themselves proved, only postulated. Scientists understand this.

"This has been succinctly summarized by saying that if a religion is something whose foundations are based on faith, then mathematics is the only religion that can prove it is a religion!"
--Wrong. Scientists understand that the provisional postulates of logic, mathematics, and the basic reliability of the human sense are just that, provisional postulates, not proved.

Scientists do not have faith in any sense. We who are scientifically minded are well aware of the provisional nature of science. It just seems to be true. I can't imagine, for example, how the principle of non-contradiction could be violated, but I cannot prove it. I take it as a working hypothesis. It is the best anybody I know of has come up with so far so I am willing to move forward on that basis until somebody demonstrates otherwise, in which case I am totally fine with adjusting my hypothesis.

"‘Godel proved that in mathematics the whole is always greater than the sum of the parts’"
--Nonsense. We simply state what seems to be the case and build up from there, perfectly well aware that the fundamental principles have not been proved, only provisionally postulated.

Miguel said...

The principle of non-contradiction as nothing more than a "working hypothesis" that could "possibly" be "violated" (even though something can only count as an actual "violation" of anything at all because of the principle of non-contradiction)? What the fuck, man.

Nonsense.

Stardusty Psyche said...

Miguel said...

" The principle of non-contradiction as nothing more than a "working hypothesis" "
--Prove it.

"that could "possibly" be "violated" (even though something can only count as an actual "violation" of anything at all because of the principle of non-contradiction)? What the fuck, man."
--Argument from incredulity.

" Nonsense. "
--Prove it.

Victor Reppert said...

How in the world would he prove it? He could show that the opposite entails a contradiction, that would beg the question, since the status of the law of noncontradiction is exactly what's at issue.

Stardusty Psyche said...

Victor Reppert said.. October 14, 2017 10:19 PM.

" How in the world would he prove it? He could show that the opposite entails a contradiction, that would beg the question, since the status of the law of noncontradiction is exactly what's at issue."
--Indeed.

It cannot be proved by any means known to any human being you or I am aware of.

Thus, it must be a provisional postulate, not something we prove, rather, something we accept provisionally absent any known or even conceivable alternative.

Legion of Logic said...

Stardusty,

It's rather curious to me that you invest so much time in these discussions in various places. I would assume that someone who doesn't really have a strong opinion on a matter would not choose to assert that opinion in debate beyond perhaps a casual chance conversation.

You have a combined belief that there are no phenomena, only illusions and approximations, and that nothing can be proven. Both of those beliefs are themselves illusive and unprovable, by your own logic, thus further undercutting the strength of any proposition you might present. All your opinions are unprovable and based on things that aren't real, according to you. Correct?

I also seem to recall that you do not believe in free will. So if we not only lack the capacity to choose, and we cannot prove anything, and anything we are discussing is not even real, why do you do this? I can't imagine believing those things and also feeling anything but futility. Illusions being helpful doesn't work, either. After all, we learned from atheists that just because a false thing brings you comfort does not mean it's acceptable to believe it.

So. We are trapped mentally, everything including ourselves is an illusion, nothing can be known, yet a thing must be known in order to be acceptable to believe, even if the unknowable truth is mentally devastating. Said devastation being an illusion, of course. And you want others to share this opinion, why exactly?

Stardusty Psyche said...

Legion of Logic said.. October 15, 2017 1:31 AM.

" All your opinions are unprovable and based on things that aren't real, according to you. Correct?"
--An approximation model of a real thing is not unreal in the sense of a pure fiction. An approximation model is considered to be valid when it converges on reality under a wide variety of circumstances.

" I also seem to recall that you do not believe in free will. So if we not only lack the capacity to choose, and we cannot prove anything, and anything we are discussing is not even real, why do you do this?"
--Because I must :-)

" I can't imagine believing those things and also feeling anything but futility."
--Yes, many people find reality depressingly futile so they console themselves with happy untruths.

Yesterday my wife gave me wrong directions when I was driving, so I turned onto the wrong street, then I had to turn around to get back on track. That would make some men angry and yell at the wife, but it simply did not bother me, in fact I found it mildly amusing.

Do you know the song When You Awake by The Band?
"You will relieve the only soul that you were born with
To grow old and never know"

In other words each of us was born, will live, die, and never know the ultimate truth of so many things. The most any of us can do is get as far as we can with the time we have and enjoy the journey while it lasts.

"I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me." --- Newton

Newton, one of the greatest known geniuses of all time, was of course correct. All his scientific discoveries were shown to not be the ultimate truth, rather, only valid approximation models.


" Illusions being helpful doesn't work, either. After all, we learned from atheists that just because a false thing brings you comfort does not mean it's acceptable to believe it."
--But it is acceptable to make use of a valid approximation while bearing in mind that the precise truth remains unknown. We have no other means available to function. If one insists on precise truth before proceeding one will become paralyzed.

" So. We are trapped mentally, everything including ourselves is an illusion, nothing can be known, yet a thing must be known in order to be acceptable to believe, even if the unknowable truth is mentally devastating. "
--I don't have any magic words to give you to avoid mental devastation but somehow I and other reductionist atheists have managed to do so. I could offer some trite platitudes like peace comes from within, and to end human suffering one must end unattainable desire. If that helps at all, great, but I doubt a bit of Buddhism will quickly help, although it might in the long term.

"Said devastation being an illusion, of course. And you want others to share this opinion, why exactly?"
--Because it is pre-determined :-)

Miguel said...

Your epistemological view of "approximation models" and complete postulates without any support whatsoever wouldn't even work or make sense without the principle of non-contradiction as *self-evident* somewhere in the top (along with other principles). Your view is confused and makes absoutely no sense, you can't even talk of objective probabilities like that, because such probabilitiws attach only because of objective tendencies in things; you can't even have probability or anything else wihout PNC as self-evident (along with other principles like PSR etc).

It should be clear to anyone who has a mind that if everything is based on unjustified or circular postulates, there could be no knowledge, and no valid "approximation models" either; to talk of probabilities or approximations would make no sense.

Your view is horribly confused and makes no sense. That's why no one agrees with you.

Stardusty Psyche said...

Miguel said.. October 15, 2017 4:59 PM.

" Your view is horribly confused and makes no sense. That's why no one agrees with you."
--Your strawman of my view is indeed confused.

Hal said...

SP,
Scientists understand that the provisional postulates of logic, mathematics,...

Those aren't postulates. They are norms of representation. Without the rules of logic and mathematics science would be incapable of making scientific discoveries or gaining new knowledge of the world.

Stardusty Psyche said...

Hal said.. October 16, 2017 5:20 PM.

SP,
Scientists understand that the provisional postulates of logic, mathematics,...

" Those aren't postulates."
--Of course they are. You can call them axioms if you prefer. The meaning is the same.

Any competent philosopher of mathematics will tell you that the whole of mathematics rests upon axioms. I prefer the word postulate because I think it expresses the aspect of the axioms I wish to emphasize in this context. I tack on the redundant qualifier "provisional" for further emphasis.

" They are norms of representation."
--Norms are not proved, merely asserted by the majority.

" Without the rules of logic and mathematics science would be incapable of making scientific discoveries or gaining new knowledge of the world."
--And such knowledge is necessarily provisional because it rests upon provisional postulates.

Maybe now you can appreciate my choice of words, "provisional postulate", as opposed to "axiom". The word "axiom" just seems to have a soft aspect that allows for notions lacking in depth of hard reasoning, such as you have expressed.

Using the term "provisional postulate" sort of slams the point all up on one's face.

Logic and math rest upon provisional postulates. Thus, all knowledge gained by logic and math is provisional. All scientific knowledge is provisional. Science doesn't do proof in the absolute sense.

To say X is scientifically proved is to necessarily say X has been provisionally demonstrated.

That all seems to bother many people very deeply. It just does not bother me.

Hal said...

Of course they are. You can call them axioms if you prefer. The meaning is the same.


I'm not calling them axioms. I'm calling them rules. Without those rules you could not make the claims you are making.

You are correct that the empirical findings of science are provisional. New knowledge may be gained that will nullify earlier findings. But that is only possible because scientists rely on those same rules of logic and mathematics to represent the new knowledge they have gained of the world.

The rules of mathematics and of logic don't change when science discovers new things. Nor do they change when scientific theories are proven false. Quite the reverse - it is the use of those very same rules that enables scientist to correct and refine their models of the world.

It's like playing a game of chess. One player may have more knowledge of chess strategies and tactics than the player he defeats but he still plays by the same rules that the losing player uses. The rules of chess don't change, but better ways may be discovered of winning the game.

Hal said...

Using the term "provisional postulate" sort of slams the point all up on one's face.

It simply strikes me as being rather flatulent. But whatever rocks your boat, go for it.

David Brightly said...

Lennox's argument seems to rest on an analogy between Formalism in maths and Ontological Reductionism in the sciences. If we take OR to be the claim that the properties of wholes can be explained in terms of the properties, the modes of interaction, and the ways of assembling simpler parts, then I suppose we might see Formalism as the idea that proofs are assemblages of axioms and rules of inference, so there is a rather rough analogy. An analogous incompleteness result in the physical sciences would have to show that there was some structure that could not be explained in this way. Gödel does this for arithmetic by representing the formulae of arithmetic with numbers themselves. But what would be the analogue of Gödel numbering in the physical sciences?

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