Our intuitions about the existence and nature of a continuum arise from perceptions of space and time [21]. But the existence of a fundamental Planck length suggests that spacetime may not be a continuum. In that case, our intuitions originate from something (an idealization) that is not actu ally realized in Nature.I would go further and say that probability does not exist in the physical world.Quantum mechanics is formulated using continuous structures such as Hilbert space and a smoothly varying wavefunction, incorporating complex numbers of arbitrary precision. However beautiful these structures may be, it is possible that they are idealizations that do not exist in the physical world.
It may come as a surprise to physicists that infinity and the continuum are even today the subject of debate in mathematics and the philosophy of mathematics. Some mathematicians, called finitists, accept only finite mathematical objects and procedures [25]. The fact that physics does not require infinity or a continuum is an important empirical input to the debate over finitism.Yes, but it is hard to prove much unless you assume mathematical infinities.
It is important to realize that the infinities are mathematical abstractions, and natural observations are all finite.
There was a concerted effort beginning in the 20th century to place infinity and the continuum on a rigorous foundation using logic and set theory. However, these efforts have not been successful. For example, the standard axioms of Zermelo-Fraenkel (ZFC) set theory applied to infinite sets lead to many counterintuitive results such as the Banach-Tarski Paradox: given any two solid objects, the cut pieces of either one can be reassembled into the other [23].No, this is wrong. ZFC is a perfectly foundation for mathematics, and is widely accepted. Those Banach-Tarski subsets are not measurable, and do not undermine ZFC.
Post-Godel there is no general agreement as to what is meant by "rigorous foundations"...No, this is a common misconception. Mathematics was on shaky foundations in the 1800s. Basic concepts like real numbers and sets had not been rigorously defined. Goedel helped show that first order logic had the properties that mathematicians needed, and helped prove that axiomatizations of set theory could be used for foundations. Soon mathematician settled on ZFC as a suitable foundation for all of mathematics.
Dear Roger,
ReplyDeleteThanks for pointing out. Interesting topic. I considered writing a blog post about it. But it would require a separate study of the papers and all. I am not enthusiastic about it, at the moment.
So, I will just mention a few points, by way of an indication.
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You say:
>> "I would go further and say that probability does not exist in the physical world." <<
Excellent observation! Also, very relevant, especially in connection with certain interpretations of QM.
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You say:
>> "Yes, but it is hard to prove much unless you assume mathematical infinities.
It is important to realize that the infinities are mathematical abstractions, and natural observations are all finite." <<
Very good point. I agree.
However, it would be necessary to tie the relevance of this *mathematical* concept of infinity to physics proper. Please see near the end of this reply.
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Hsu says (in conclusions, IV. FINITISM):
>> "Physics – i.e., models which can be compared to experimental observation, actual “effective procedures” – does not ever require infinity, although it may be of some conceptual convenience." <<
No, if the suggestion is that physics is all (or mainly) about models / modelling, then I don't accept it. Modelling is useful in physics (and it plays an extremely useful part in my own theory). But models is not the ultimate aim of physics. The ultimate aim is: physical laws and theories that are *general* enough, i.e., knowledge.
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Hsu says (in his blog post, ICML notes):
>> "This "it" (mathematics) that Cohen describes may be the set of idealizations constructed by our brains extrapolating from physical reality." <<
Ah... So, there. That's source of the troubles.
No, the issue isn't as narrow as just "idealizations", say of physical objects, or of their attributes or actions, e.g., studying the collision of a cricket ball with a bat, by idealizing it as a perfectly rigid sphere. The issue is as big as *concepts*.
No, it's enough to say that concepts are constructed by brains. That will be tantamount to leaving a vital essential: *Concepts* are (volitionally) *conceived* by the *mind*. The more automatic cognitive functions of the body are relevant only at the perceptual level. For instance, the eye and the brain work together in such a way that we perceive objects and not just unrelated blobs of color. But it's the mind---the non-material aspect of man, his consciousness---which is vital to concept formation.
No, the process isn't one of extrapolation. Notice, extrapolation from a sequence of *concrete* objects would keep you at the same level, viz., that of more concrete objects. Or at the level of your perceptions of them. But *extrapolation* won't give you concepts. The process is one of concept formation.
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Now, one last point. I will just mention it, without tieing it with the rest.
The infinity, as you pointed out, *is* a *mathematical* concept. The relevance of this mathematical concept to physics is this:
Since Newton and Leibniz, physical laws (and quite often, also the physical quantities i.e. variables used in them) are formulated using the differential equations paradigm. At the base of this paradigm is the notion of limits. And the limiting processes are infinite processes.
In other words, the use of infinity helps explicitize how the physicist reaches the level of abstractions, in defining his crucial concepts. The infinity is the physicist's *means* of abstraction. Infinity is *that* crucial to physics---at least for physics since the invention of the calculus. And hence, it's also absolutely *indispensable* to physics.
Best,
--Ajit
Typo:
Delete"No, it's enough to say that concepts are constructed by brains."
-->
"No, it's *not* enough to say that concepts are constructed by brains."
Best,
--Ajit
You can't do anything forever. Even math. Math much like music requires it to be performed, it doesn't happen in a meaningless timeless instant or over an equally ridiculous eternity.
ReplyDeleteMath is a logical process, it follows rules that must be applied in certain sequences in order to be correct and remotely useful. Every algorithm is at heart:
" n.
A finite set of unambiguous instructions that, given some set of initial conditions, can be performed in a prescribed sequence to achieve a certain goal and that has a recognizable set of end conditions. "
Notice the words "finite", "unambiguous", "prescribed sequence" and "initial conditions".
ALL math requires time to even be an 'abstraction'.
There are NO instantaneous calculations even in consideration, as in all of them, there is process, and any process no matter how relatively small still requires finite time.
If you want to quibble about abstract taffy like limits, consider finite calculus instead. Nothing lasts forever, check the obituaries, even mathematicians are dropping like flies.
You have limited time, now act like it and leave infinity and infintesimals to the metaphysicians who have nothing particularly useful or urgent to do.