TA.770. In Our Mathematical Universe -- My Quest for the Ultimate Nature of Reality (Alfred A. Knopf/Random House 2014), the MIT physicist Max Tegmark finds that reality IS a mathematical structure. That is, math not only describes reality, math is reality. (I suppose that the concept of mathematics replaces the former concept of universal spirit.)
According to Tegmark,
But, is the cosmos a set of numbers only? The purpose of the bulk of physics is to provide mathematical relations for phenomena in order to predict future phenomena. The human brain determines -- analyzes (i.e. cuts up) -- wholes in order to identify phenomena that might be useful to be aware of.
Tegmark in his book discusses the peculiar mathematical necessity of the relativistic spacetime block, in which the "flow of time," space with a reference frame, motion, velocity and energy all vanish into a frozen four-dimensional continuum. So, I guess Tegmark is saying that if time, energy and other physical phenomena don't really exist, then the reality of the spacetime block must be "pure mathematics." Of course, he knows that physics does not now have a "theory of everything" that potentially exactly mirrors all phenomena. No one has managed to unite relativity and quantum theory.
Tegmark embraces a multiverse scheme of the cosmos, and sees that picture as entirely mathematical.
I see that notion as problematic. It's a bit like saying that if you had a really, really, really accurate computer simulation of an approaching storm, the simulation and the storm would merge into one. In fact, that thought isn't absurd, but such a simulation would require extremes of precision that are unattainable.
How so? At that level of precision, quantum effects become important. This means quantum effects in the simulation would bring about divergences, some of which take on a chaos character. Similarly, Werner Heisenberg teaches us that we cannot tell the computer the exact spacetime point of any particle. That's because, it is in principle impossible to measure both the momentum (or velocity) and the position precisely. Now would this quantum of uncertainty be so influential as to cause major divergences between prediction and actual behavior? I would say so, since we have the impact of trillions of quanta of uncertainty. Some "uncertainty collectives" are liable to drive substantial divergences.
Still, Tegmark is not talking about simulations but about some precise mathematical theory that one day covers all physics. Nevertheless, spacetime block or no, any physical theory must include rules for computing what happens when X is applied. And each such computation can, most agree, in principle formulated as a Turing machine (algorithm). Now it is true that Alan Turing did not take into account quantum fluctuations in his brilliant paper on the halting problem. But for Tegmark's scheme to work, we would need to take those under consideration -- in which case the scheme is inconsistent.
But, it should be pointed out, that Tegmark was giving a popular exposition intelligible to bright high school students. I have not read his academic papers, which are likely far more nuanced.
But even if we grant that the TM issue is a red herring, I still have difficulty with the idea that, as long as there is an exact match, a set and what it describes are necessarily equivalent -- in fact, identical. A mathematical set is a mental abstraction. If Tegmark means to say that the cosmos is likewise a mental abstraction, one would think he would have brought some sort of idealism or straightforward theism into his argument, but that was far from his mind, obviously.
An obvious example: the set {Ø, {Ø}}, (which reduces to {{Ø}} ), is defined as representing the abstract concept we call the number 1. We may designate a basket with a lone apple as a set (but not a mathematical set) if we wish. In that case, the mathematical set named "1" is found to be in one-to-one correspondence with the members of the basket set. In that case, convention rules that we say the basket set has "1" apple in it.
But the basket and apple do not represent a "pure set." Only sets built up from a logic theory and a set theory that coherently represent number systems are ordinarily the basis for mathematical sets. Now then, the cosmic physical reality considered as a giant basket that is exactly represented by a pure set is not itself a pure set. But, Tegmark says there is a case for that cosmic basket being a pure set because, I suppose, in manifolds of four or more dimensions, action ceases. Everything is theoretically static. Yet it is hard to credit the idea that if someone one day writes down the governing equation or equations for a "theory of everything," that they contain the potential for an exact point-by-point spacetime-plus description of the cosmos AND that the potential cosmos ITSELF is implicit in this or these equations.
In Tegmark's favor, we find that Stephen Wolfram, a mathematician and physicist, has similar beliefs:
According to Tegmark,
Remember that two mathematical structures are equivalent if you can pair up their entities in a way that preserves all relations. If you can thus pair up every entity in our external physical reality with a corresponding one in a mathematical structure ("This number here in physical space corresponds to this number in the mathematical structure, for example), then our external physical reality meets the definition of being a mathematical structure -- indeed, that same mathematical structure.That is, we can examine two sets of (mathematical) relations and see whether each description aRb in R is matched one-to-one by another description xSy in S. By "match," we mean any aRb is replicated by exactly one xSy. From standard set theory, we know that when two sets are equivalent, they are identical.
But, is the cosmos a set of numbers only? The purpose of the bulk of physics is to provide mathematical relations for phenomena in order to predict future phenomena. The human brain determines -- analyzes (i.e. cuts up) -- wholes in order to identify phenomena that might be useful to be aware of.
Tegmark in his book discusses the peculiar mathematical necessity of the relativistic spacetime block, in which the "flow of time," space with a reference frame, motion, velocity and energy all vanish into a frozen four-dimensional continuum. So, I guess Tegmark is saying that if time, energy and other physical phenomena don't really exist, then the reality of the spacetime block must be "pure mathematics." Of course, he knows that physics does not now have a "theory of everything" that potentially exactly mirrors all phenomena. No one has managed to unite relativity and quantum theory.
Tegmark embraces a multiverse scheme of the cosmos, and sees that picture as entirely mathematical.
I see that notion as problematic. It's a bit like saying that if you had a really, really, really accurate computer simulation of an approaching storm, the simulation and the storm would merge into one. In fact, that thought isn't absurd, but such a simulation would require extremes of precision that are unattainable.
How so? At that level of precision, quantum effects become important. This means quantum effects in the simulation would bring about divergences, some of which take on a chaos character. Similarly, Werner Heisenberg teaches us that we cannot tell the computer the exact spacetime point of any particle. That's because, it is in principle impossible to measure both the momentum (or velocity) and the position precisely. Now would this quantum of uncertainty be so influential as to cause major divergences between prediction and actual behavior? I would say so, since we have the impact of trillions of quanta of uncertainty. Some "uncertainty collectives" are liable to drive substantial divergences.
Still, Tegmark is not talking about simulations but about some precise mathematical theory that one day covers all physics. Nevertheless, spacetime block or no, any physical theory must include rules for computing what happens when X is applied. And each such computation can, most agree, in principle formulated as a Turing machine (algorithm). Now it is true that Alan Turing did not take into account quantum fluctuations in his brilliant paper on the halting problem. But for Tegmark's scheme to work, we would need to take those under consideration -- in which case the scheme is inconsistent.
But, it should be pointed out, that Tegmark was giving a popular exposition intelligible to bright high school students. I have not read his academic papers, which are likely far more nuanced.
But even if we grant that the TM issue is a red herring, I still have difficulty with the idea that, as long as there is an exact match, a set and what it describes are necessarily equivalent -- in fact, identical. A mathematical set is a mental abstraction. If Tegmark means to say that the cosmos is likewise a mental abstraction, one would think he would have brought some sort of idealism or straightforward theism into his argument, but that was far from his mind, obviously.
An obvious example: the set {Ø, {Ø}}, (which reduces to {{Ø}} ), is defined as representing the abstract concept we call the number 1. We may designate a basket with a lone apple as a set (but not a mathematical set) if we wish. In that case, the mathematical set named "1" is found to be in one-to-one correspondence with the members of the basket set. In that case, convention rules that we say the basket set has "1" apple in it.
But the basket and apple do not represent a "pure set." Only sets built up from a logic theory and a set theory that coherently represent number systems are ordinarily the basis for mathematical sets. Now then, the cosmic physical reality considered as a giant basket that is exactly represented by a pure set is not itself a pure set. But, Tegmark says there is a case for that cosmic basket being a pure set because, I suppose, in manifolds of four or more dimensions, action ceases. Everything is theoretically static. Yet it is hard to credit the idea that if someone one day writes down the governing equation or equations for a "theory of everything," that they contain the potential for an exact point-by-point spacetime-plus description of the cosmos AND that the potential cosmos ITSELF is implicit in this or these equations.
In Tegmark's favor, we find that Stephen Wolfram, a mathematician and physicist, has similar beliefs:
Underneath, [cosmic space is] a bunch of discrete, abstract relations between abstract points. But at the scale we're experiencing it, the pattern of relations [that space] has makes it seem like continuous space of the kind we're used to. It's a bit like what happens with, say, water. Underneath, it's a bunch of discrete molecules bouncing around. But to us, it seems like a discrete fluid.A Project to Find the Fundamental Theory of Physics
by Stephen Wolfram
(Wolfram Media 2020) From what I can gather -- having only scanned that book -- Wolfram believes that we only need find a few simple rules in order to unlock the enigma of existence.
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