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I recently watched this interesting interview with physicist Paul Davies. https://www.youtube.com/watch?v=vqZN_LGYHJc

In the first couple of minutes he outlines some of the problems with a multiverse where "all possible universes" exist. He says that by "all possible universes" mathematical physicists tend to mean all universes which are mathematically describable, but he suggests that this is somewhat arbitrary. Why stop at mathematics? Why not allow for all possible universes which are aesthetically or morally possible?

It seems to me that an aesthetically or morally possible universe would need to logically possible in order to exist. So my question is this: Can there be an aesthetically or morally possible universe (which is also logically possible) which is not mathematically possible, or cannot be described mathematically?

My understanding is that if something is logically possible, it is mathematically possible. Is this incorrect?

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  • It's fairly clear what mathematical possibility amounts to. But what would aesthetic or moral possibility amount to? I'm not sure I can make sense of that. – Eliran Apr 2 '16 at 20:31
  • It is in some sense logical that no one should starve when others have enough. But it is not always mathematically possible to ensure everyone has adequate calories to live. So it seems to me that the answer is definitely 'no'. – user9166 Apr 4 '16 at 23:24
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Davies does not give a definition of aesthetically or morally possible universe.

One can doubt whether such concepts are helpful, because neither aesthetics nor moral are properties of our own universe qua universe. Both are properties which require humans and can only be defined relative to them. E.g., recall the saying "Beauty arises in the mind's eye of the discerning beholder". And morality is a set of values and rules established in a society.

On the opposite, a mathematical universe is a universe the laws of which can be formalized by mathematics. E.g., the natural laws of our own universe can be captured in many cases by differential equations. But a universe which is completely governed by chance cannot be described by mathematical rules. Hence not every possible universe must be a mathematical universe.

We cannot imagine or describe a universe where the rules of logic are no longer valid, e.g., a universe where the law of non-contradiction never holds. The minimal requirement of a possible universe is the validity of our logic.

As a consequence, a universe which is logically possible is not necessarily a mathematical universe.

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  • Thank you, that helps clear up my thinking! I hadn't thought of universes governed entirely by chance. Although it seems difficult to imagine such a universe, I guess there wouldn't need to be any inherent contradiction within such a universe, at least if we limit ourselves to laws describable by differential equations. I suppose someone could argue that if a universe is non-empty, then it contains things which can be enumerated, and thus that universe has mathematical properties and is mathematically describable. – M47145 Apr 2 '16 at 20:16
  • @M47145 As an analogue to a universe completely governed by chance I often conceive clouds in the sky. Beings who live in the clouds of the sky would experience their environment highly unstructured. It is a fuzzy world. No computation is possible because of the chaotic character of the laws. There are laws and they are deterministic. But due to their chaotic character - sensible dependency on initial conditions - this universe appears to its inhabitants like a universe governed by chance. – Jo Wehler Apr 2 '16 at 20:36
  • Hmm that analogy doesn't seem helpful to me if I'm understanding it correctly, because even then it seems nothing would prevent the beings who live in the cloud from potentially coming up with the theory of brownian motion, chaos theory, or mathematical models to describe stochastic processes etc. This of course would make such a cloud universe mathematically describable using differential equations. – M47145 Apr 2 '16 at 21:05
  • @M47145 I consider it impossible to derive soley from the observation of the chaotic behaviour in the cloud the underlying differential equation. - Please note that I termed the cloud environment an analogue: We from the outside know the differential equations of meteorology. But for the observer in the cloud this environment seems governed by chance. – Jo Wehler Apr 2 '16 at 22:29
  • I would contend that, given Russel's paradox and other aspects, ours is not a universe which is "logically possible". Our logic is an approximation to reality, which actually fails internally, so the minimal requirement for possibility cannot involve agreeing with our logic. Expecting such is somewhere between "putting the cart before the horse" and "taking the map as the territory". – user9166 Apr 6 '16 at 16:31
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I think that we constantly imagine moral universes that are not logical. Our legal systems represent such things, and they work hard to attain logical consistency through continual refinement, yet they evolve more internal contradictions all the time.

We generally imagine that we can state a system will all the constraints we would like to make us morally satisfied, and work out the conflicts later. But the potential conflicts are always essentially omitted from the concept of the system. In fact, the actions that render the morally compelling system logical and tractable arise ad hoc, and seldom resolve the actual logical problem causing the conflict until there are many, many instances of the same kind of compromise between moral principles.

I think that legal systems, and therefore the moralities they attempt to approximate, present a paraconsistent logic with only a local version of the law of non-contradiction in the same way that Intuitionism and other constructive mathematics present logics with only a local version of the law of the excluded middle (which applies, in those contexts, only when the options have been reduced to a finite number, or have been crowded in by proofs on all sides that reduce the problem to something essentially finite.)

So I would suggest that these two kinds of imagination converge on a single notion of sound human logic from opposite directions and that neither is a reasonable requirement to enforce on the other. We should not expect morality to be tractable, or mathematics to be humane. We need to live with systems that work despite not being complete in either way.

I would also propose that both of these layers of partial reasoning are forms of aesthetics. Mathematics is based upon its own feelings of consistency and clarity, sometimes referred to as essential elegance, and morality is based upon its own feelings of propriety and order, sometimes referred to as essential humanity.

So perhaps only the set of aesthetically appealing universes with a given set of interacting sources of value is really a good model. It can capture these two, and other human drives.

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No offense, but I think Jo Wehler's answer is profoundly wrong. "We cannot imagine X" really means "I cannot imagine X". And "We cannot imagine X, therefore X is not possible" is obviously absurd. And what justifies the implied proposition that moral and aesthetic properties "require humans" but mathematical ones don't? Ever met a non-human mathematician? This kind of argument is just a coded way of saying that mathematicians have special, trans-historical, inter-galactic epistemological authority, which is patently false.

Mathematics and logic are hisorical, just like every other form of discourse. What counts as true, or possible, or necessary changes. 300 years ago, it was impossible to imagine lots of stuff, like relativity and non-euclidean geometry, that is now accepted as just obvious.

To the original question, I confess I have no idea what phrases like "mathematically possible" or "logically possible" even mean. Both are usually conducted in the indicative mood, so to speak. Modal concepts like "possible" and "necessary" are not a part of traditional mathematical and logical discourse, as far as I can tell. What would "possible, but only mathematically" mean? Doesn't "logically possible" just mean "expressible in one of the logical formalisms we currently like? There are lots of things we cannot express in any of our formalisms. ("There are more things in heaven and earth, Horatio,...") Why should an ethically or aesthetically possible universe not feel free to ignore our paltry mathematical and logical concepts?

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  • Thanks for your answer! I would argue that although mathematics and logic are historical forms of discourse, they are our descriptions of an underlying reality. Relativity as you say, held 300 years ago, regardless of whether people knew that it did. There are things which logical formalism cannot currently (or ever) describe, however, that doesn't mean those things are illogical. I understand a logically impossible universe to be one in which there exists married bachelors, square circles, or other logically impossible concepts. – M47145 Apr 4 '16 at 0:18
  • Perhaps one should try to think how people would have answered before Non-euclidean geometry was discovered. And next to consider if mathematics is reductible to logic (with a passing thought about incomputable numbers)... – sand1 Apr 4 '16 at 8:29
  • @mobileink Please read the whole sentences from my answer: 1) We cannot imagine or describe a universe where the rules of logic are no longer valid 2) Both are properties which require humans and can only be defined relative to them. - On the base of correct quotation I would enjoy to discuss our different views. – Jo Wehler Apr 4 '16 at 19:41
  • Most folks don't think math is historically determined. It is not a discourse, but a set of standards for allowed discourses. If we discover physics that requires new math, that math is already math, even though it has not been worked out yet... A mathematically possible/consistent universe is one that could have a mathematical structure put behind it, not one that already has one, or our own universe would only become mathematically possible once physics ended. – user9166 Apr 6 '16 at 16:41
  • "our descriptions of an underlying reality" - It's not just the language that is historic (contingent), its the concepts themselves. The concept "number" has changed drastically over the centuries. What's the "underlying reality" of i (sqrt of -1)? I guess my response to your original question is that the relation between logic (revolving around the central notion of valid inference) and ontology is murky at best. The problem with "married bachelor" is material, not logical; if there are no married bachelors it is not because the concept is illogical but because it is materially incoherent. – user20153 Apr 7 '16 at 15:11

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