In “Ultimate logic: To infinity and beyond” ( New Scientist, 01 August 2011) Richard Elwes tells us,
This curiously intractable riddle is known as the continuum hypothesis, and it concerns that most enigmatic quantity, infinity. Now, 140 years after the problem was formulated, a respected US mathematician believes he has cracked it. What’s more, he claims to have arrived at the solution not by using mathematics as we know it, but by building a new, radically stronger logical structure: a structure he dubs “ultimate L”.
For most purposes, life within these structures is the same: most everyday mathematics does not differ between them, and nor do the laws of physics. But the existence of this mathematical “multiverse” also seemed to dash any notion of ever getting to grips with the continuum hypothesis. As Cohen was able to show, in some logically possible worlds the hypothesis is true and there is no intermediate level of infinity between the countable and the continuum; in others, there is one; in still others, there are infinitely many. With mathematical logic as we know it, there is simply no way of finding out which sort of world we occupy.
That’s where Hugh Woodin of the University of California, Berkeley, has a suggestion. The answer, he says, can be found by stepping outside our conventional mathematical world and moving on to a higher plane.
What is it about the expression “higher plane” that sets the horses bolting?
Before we bid farewell to meaningful numbers, thoughts?
9 Replies to “A radically stronger logical structure than mathematics?”
In a comment on an earlier thread, I stated my view that mathematics is invented rather than discovered.
Mathematical Platonists, who tend to think that mathematics is metaphysical, usually take the continuum hypothesis to be a genuine unsolved problem. Mathematical fictionalists (such as myself), who tend to see mathematics as a human construct, are more likely to view the continuum hypothesis as a pseudo-problem.
As for “higher plane”, most mathematicians would probably think that a familiar expression.
How come this ‘radically stronger logical structure’ of a ‘mathematical “multiverse”’ reminds me, somewhat, of the ontological argument?
Ontological Argument For God From The Many Worlds Hypothesis – William Lane Craig – video
God Is Not Dead Yet – William Lane Craig – Page 4
The ontological argument. Anselm’s famous argument has been reformulated and defended by Alvin Plantinga, Robert Maydole, Brian Leftow, and others. God, Anselm observes, is by definition the greatest being conceivable. If you could conceive of anything greater than God, then that would be God. Thus, God is the greatest conceivable being, a maximally great being. So what would such a being be like? He would be all-powerful, all-knowing, and all-good, and he would exist in every logically possible world. But then we can argue:
1. It is possible that a maximally great being (God) exists.
2. If it is possible that a maximally great being exists, then a maximally great being exists in some possible world.
3. If a maximally great being exists in some possible world, then it exists in every possible world.
4. If a maximally great being exists in every possible world, then it exists in the actual world.
5. Therefore, a maximally great being exists in the actual world.
6. Therefore, a maximally great being exists.
7. Therefore, God exists.
Now it might be a surprise to learn that steps 2–7 of this argument are relatively uncontroversial. Most philosophers would agree that if God’s existence is even possible, then he must exist. So the whole question is: Is God’s existence possible? The atheist has to maintain that it’s impossible that God exists. He has to say that the concept of God is incoherent, like the concept of a married bachelor or a round square. But the problem is that the concept of God just doesn’t appear to be incoherent in that way. The idea of a being which is all-powerful, all knowing, and all-good in every possible world seems perfectly coherent. And so long as God’s existence is even possible, it follows that God must exist.
I like the following concluding comment about the ontological argument from the Dr. Plantinga video:
“God then is the Being that couldn’t possibly not exit.”
Ontological Argument – Dr. Plantinga (3:50 minute mark)
Finely Tuned Big Bang, Elvis In The Multiverse, and the Schroedinger Equation – Granville Sewell – audio
At the 4:00 minute mark of the preceding audio, Dr. Sewell comments on the ‘transcendent’ and ‘constant’ Schroedinger’s Equation;
‘In chapter 2, I talk at some length on the Schroedinger Equation which is called the fundamental equation of chemistry. It’s the equation that governs the behavior of the basic atomic particles subject to the basic forces of physics. This equation is a partial differential equation with a complex valued solution. By complex valued I don’t mean complicated, I mean involving solutions that are complex numbers, a+b^i, which is extraordinary that the governing equation, basic equation, of physics, of chemistry, is a partial differential equation with complex valued solutions. There is absolutely no reason why the basic particles should obey such a equation that I can think of except that it results in elements and chemical compounds with extremely rich and useful chemical properties. In fact I don’t think anyone familiar with quantum mechanics would believe that we’re ever going to find a reason why it should obey such an equation, they just do! So we have this basic, really elegant mathematical equation, partial differential equation, which is my field of expertise, that governs the most basic particles of nature and there is absolutely no reason why, anyone knows of, why it does, it just does. British physicist Sir James Jeans said “From the intrinsic evidence of His creation, the great architect of the universe begins to appear as a pure mathematician”, so God is a mathematician to’.
i.e. the Materialist is at a complete loss to explain why this should be so, whereas the Christian Theist presupposes such ‘transcendent’ control,,,
In the beginning was the Word, and the Word was with God, and the Word was God.
as to this statement:
and this staement;
Perhaps there is a way to know if there is a ‘intermediate level of infinity between the countable and the continuum’ in the universe we live in:
The End Of Christianity – Finding a Good God in an Evil World – Pg.31
William Dembski PhD. Mathematics
Excerpt: “In mathematics there are two ways to go to infinity. One is to grow large without measure. The other is to form a fraction in which the denominator goes to zero. The Cross is a path of humility in which the infinite God becomes finite and then contracts to zero, only to resurrect and thereby unite a finite humanity within a newfound infinity.”
The Center Of The Universe Is Life – General Relativity, Quantum Mechanics, Entropy and The Shroud Of Turin – video
The conflict of reconciling General Relativity and Quantum Mechanics appears to arise from the inability of either theory to successfully deal with the Zero/Infinity problem that crops up in different places of each theory:
THE MYSTERIOUS ZERO/INFINITY
Excerpt: The biggest challenge to today’s physicists is how to reconcile general relativity and quantum mechanics. However, these two pillars of modern science were bound to be incompatible. “The universe of general relativity is a smooth rubber sheet. It is continuous and flowing, never sharp, never pointy. Quantum mechanics, on the other hand, describes a jerky and discontinuous universe. What the two theories have in common – and what they clash over – is zero.”,, “The infinite zero of a black hole — mass crammed into zero space, curving space infinitely — punches a hole in the smooth rubber sheet. The equations of general relativity cannot deal with the sharpness of zero. In a black hole, space and time are meaningless.”,, “Quantum mechanics has a similar problem, a problem related to the zero-point energy. The laws of quantum mechanics treat particles such as the electron as points; that is, they take up no space at all. The electron is a zero-dimensional object,,, According to the rules of quantum mechanics, the zero-dimensional electron has infinite mass and infinite charge.
Quantum Mechanics and Relativity – The Collapse Of Physics? – video – with notes as to plausible reconciliation that is missed by materialists
The following Professor offers a very interesting insight into this issue of ‘reconciling’ the ‘mental’ universe of Quantum Mechanics with the space-time of General Relativity:
How the Power of Intention Alters Matter – Dr. William A. Tiller
Excerpt: “Most people think that the matter is empty, but for internal self consistency of quantum mechanics and relativity theory, there is required to be the equivalent of 10 to 94 grams of mass energy, each gram being E=MC2 kind of energy. Now, that’s a huge number, but what does it mean practically? Practically, if I can assume that the universe is flat, and more and more astronomical data is showing that it’s pretty darn flat, if I can assume that, then if I take the volume or take the vacuum within a single hydrogen atom, that’s about 10 to the minus 23 cubic centimeters. If I take that amount of vacuum and I take the latent energy in that, there is a trillion times more energy there than in all of the mass of all of the stars and all of the planets out to 20 billion light-years. That’s big, that’s big. And if consciousness allows you to control even a small fraction of that, creating a big bang is no problem.” – Dr. William Tiller – has been a professor at Stanford U. in the Department of materials science & Engineering
I would hold General Relativity to be a ‘countable’ infinity:
through a fairly exhaustive examination of the General Relativity equations themselves, acknowledges the insufficiency of General Relativity to account for the ‘completeness’ of 4D space-time within the sphere of the CMBR from different points of observation in the universe.
The Cauchy Problem In General Relativity – Igor Rodnianski
Excerpt: 2.2 Large Data Problem In General Relativity – While the result of Choquet-Bruhat and its subsequent refinements guarantee the existence and uniqueness of a (maximal) Cauchy development, they provide no information about its geodesic completeness and thus, in the language of partial differential equations, constitutes a local existence. ,,, More generally, there are a number of conditions that will guarantee the space-time will be geodesically incomplete.,,, In the language of partial differential equations this means an impossibility of a large data global existence result for all initial data in General Relativity.
I enjoyed that New Scientist article, despite not being a mathematician.
Made for an interesting extended Saturday breakfast 🙂
“Mathematics as we know it” can’t deal with ‘infinity’ … as witness that n/0 = ‘infinity’, yet this truth can never be proven within “mathematics as we know it”.
my view that mathematics is invented rather than discovered.
Do you have the same view of logic?
How much is 100 dots divided by 0 symbols?
That, of course, does not compute, any more that dividing 0 apples by 100 oranges.
But the apples and oranges could be a message from some alien intelligence, so they could be symbols, right?
The fact that there are 0 apples could be a message that aliens hate apples. Or it could be a message that they love apples, and ate them all.
So the missing apples could be information, right? We could measure the missing information by taking one byte at a time.