Math vs. The Universe

Great Debates
1

I’m continuing the discussion that evolved rather sharply from this thread started by the cryptic rwjefferson (to say the least), which was eventually closed due to rwj’s unintelligible “poetry”.

Mostly I found the discussion interesting, if a little unfocused, probably because the OP was asking if mass is really a cosmic “drain” that spacetime flows into. I think. Then something about ducky waves. :confused: Whatever…

Anyway, then** Angry Lurker** posted this thread in ATMB, asking if it could be reopened, feeling as if the baby’s been thrown out with the bathwater.

The thread did kind of cool near the end, but I wonder if a more focused discussion concerning modern mathematics in physics, QM and cosmology, and how it describes reality might warm the discussion back up.

I’ve spoke some of my mind in rwj’s thread on the topic (if not feeling slightly in over my head), and believe that math is the torch lighting the way to the secrets of the universe, but in the end, we’ll need other tools to make more sense out of it. That is, figuring our minds are capable enough to figure it all out.

So, some primer questions:

What are Math’s weaknesses and strongpoints in describing reality?

How far can our math take us into uncovering and describing some of the fundamental aspects of nature? Is it boundless?

Besides math, is there any other tools of human thought (abstract, rigorous, rhetorical, etc.) that can help us understand the universe? Might we devise new ones, like true AI?

Can math help with uncovering some of the problems concerning sentience that has arisen within the universe?

2

Mathematics is certainly essential to help achieve the eventual explanation of everything. However, I don’t believe that any final “Theory of Everything” (TOE) will ever be developed by a pure mathematician, despite their pretensions to be physicists. String theorists in particular are probably the best examples of pure mathematicians wasting research money. They predict nothing and can only try to squeeze in explanations of new discoveries, with modest success, after the event.

If a TOE is ever comes into being it will be achieved by someone who’s a physicist and observer first and a mathematician second.

A good example (to me, at any rate) is the claim regularly made by particle physicists that they have “created matter” in colliders with high energy (6.3 Gev) proton to proton collisions. This has a one in three million outcome:

P1 + p2 > p1 + p2 + p3 + (-p)1

The antiproton being (-p)1 which then goes to find a fourth proton, p4, with which to “annihilate”. Up until then p4 had been a normal proton for the last 12 billion years or so behaving as a normal proton until (-p)1 suddenly came into its life.

complete details can be found in this University of Virginia Michael Fowler article :
Transforming Energy into Mass: Particle Creation
Notwithstanding the claims made in the article, my understanding is that ‘m’ in E=mc^2 stands for ‘mass’ not ‘matter’.

Rather than involving the creation of new ‘matter’ – a highly organised proton and an equally highly organised antiproton, and always one of each - from the totally chaotic flash of short lived particles that result from proton to proton collisions, is it possible that dark matter is comprised of particle-anti particle unions that have almost no links (gravitational, etc) to the material universe to become near massless, invisible and almost undetectable rather than the existing mutual annihilation theory advanced by pure mathematician/physicists who work at these colliders?

I have addressed this question to several university physicist/mathematicians I know personally and they all say the same thing. That there is no “hidden stuff” in space according to Einstein’s version of relativity theory and that because the equations all balance out eventually (after 13 millionths of a second) everything’s fine. They won’t even consider that there could be any other possibility.

As far as I can determine, this is an example of pure mathematics (the equations all balance, don’t you see?) blocking off any possible alternative lines of inquiry about an observed physical phenomenon because, no matter how implausible it is that two highly organised particles can be “created” de novo from pure disorganised chaos, the equations on a piece of paper say it must be so and therefore it is true.

3

Perhaps slightly tangential to the discussion, but has the OP read The Unreasonable Effectiveness of Mathematics in the Natural Sciences?

(There’s also another interesting spin off from that article: The Unreasonable Effectiveness of Logic in Computer Science.)

4

I did not read the referenced thread but that won’t stop me from putting in my two cents!

Maths are languages in which scientists compose poems that attempt to describe reality. Metaphors of reality if you will. Just as poets are trying to get at essential truths so are scientists.

In the spoken world of course certain languages can be better utilized to express particular concepts and in others those same concepts are inarticulable. Certain languages lead us to express certain kinds of concepts more readily. So it is with maths.

String theory for example was the result of discovering that a different language made for some real neat rhyming schemes that seemed suddenly really useful. How great the poems created with that newer language really are, and how much we conclude they capture “truth”, upon critical review and repeated hearings is a separate debate.

5

You’re missing the middle paper in that series.

6

Here’s a more specific question that may serve to focus the discussion.

Is the universe governed by the law of non-contradiction?

7

The strong point of math is that it describes physical objects and actions remarkably well. If you know the strength of a magnetic field, you can predict the acceleration of an electron in that field with remarkable accuracy.

The main weak point of math is that it offers no way of describing things that exist at higher levels, such as happiness, satisfaction, fear, personality, sadness, freedom, fulfillment, despair, and so forth. This has become problematic in recent years because some people erroneously believe that these things can be tackled mathematically. For instance, they may ask people to rank the satisfaction they get on a numerical scale, or simply come up with an arbitrary way of ranking freedom. (There’s a thread going about such a study right now, in fact.) Such approaches will obviously lead to error, because those things have no numerical ranking. Personality has no measure. Happiness has no measure.

Another weak point of math is that many mathematicians have drifted into highly abstract fields of mathematics that have minimal relationship to reality.

8

It seems to me that logical laws are descriptive of the world.

Also, doesn’t the liar’s paradox suggest that the law of non contradiction isn’t absolute? Or at least that classical logic isn’t absolute?

9

Does answering that question actually clear up anything? Because even if it is, is that because you need the law of non-contradiction to have a universe, or is non-contradictoriness merely a property of our universe?

In principle, one could settle your question empirically, by performing an exhaustive search for contradictions within the universe (provided, of course, that the universe is exhaustively searchable, and thereby most importantly not infinite). But then, one would still have no answer on whether non-contradiction (and by extension logical principles in general) forms a supervening principle for all conceivable universes, or whether it just happened to work out that way in our case.

What I’m trying to get at (not very clearly, perhaps) is that the apparent mathematical underpinnings of our universe would be a great deal less mysterious if they were, in some way, a result of necessity rather than chance. For, while it is probably not terribly surprising that there exist languages that allow us to express a great deal of ideas about the state of the world – such things can also be achieved in natural languages, after all --, it does seem a bit surprising that there is one that seems so explicitly streamlined just for this purpose, while still being relatively easy and understandable.

So, to determine whether it is logic and mathematics that give the universe its shape, or if that shape merely gives rise to the notions encapsulated in those frameworks, is a question of greater interest IMO.

10

Does the law of non-contradiction describe how things are, or how things must be?

Some people use the Liar to argue for a third truth value that is neither True nor False. Other people use it to argue that not all sentences are truth-value-evaluable. I think I remember reading someone’s approach that involved saying the Liar’s sentence is actually ambiguous between the members of an infinite class of sentences, half of which are true and half of which are false. (Maybe I conjured that one up in a dream.) And there are other approaches to take besides.

11

As someone who does such things for a living, I must disagree.

It is inherently much fuzzier then ultra-precise physics…but you CAN measure happiness among other fuzzy things.

12

Isn’t this basically the Anthropic Principle?

13

I don’t know, but I wasn’t hoping to clear up anything. Just to “focus discussion.” :wink:

See my previous post. I ask there whether the law of non-contradiction describes the way the universe is, or the way it must be–assuming it describes the universe at all.

I wonder if it’s possible to know one has searched for every possible violation of non-cont. even in a finite universe. Even if there are only a finite number of facts, might it be that the truth of the fact “those are all the facts” is inaccessible?

Huh. I’d find it really mysterious if our universe is governed by non-cont. because non-cont. is in some way “necessary” in a way that is logically prior to the existence of the things in the universe. Where does such necessity come from?!

Whether non-cont. is metaphysically necessary or not, you’re touching on something I find surprising either way. A lot of people want to argue that non-cont. isn’t a rule governing the universe, but rather governing our languages. But I’m amazed if this rule governing our language, (a rule developed probably as a result of evolution by natural selection in a way depending very much on highly contingent properties of our local environment,) somehow also governs events taking place everywhere in the visible universe and beyond!

That is, I’m surprised unless this particular fact about our environment–that it is governed by non-cont., isn’t just true locally but really is true globally. (I feel certain that it is, of course. But should I?) But then, that just invites my question. Is it true globally? If it is, is this because the universe is governed by the rule, in some sense implying necessity? Or is it a “contingent” fact about the universe, something about how the universe “just happens” to be? If the universe is governed by the rule in a way that implies necessity, where does the necessity come from? If it’s how the universe “just happens” to be, does this mean it really could have been otherwise? If so, why wasn’t it? What explains the odd coincidence that the rule just happens to be accurate at each of the huge number of points existing in our universe?

In fact I thought my question was a very direct way of asking your question.

14

Here’s what’s possible, though. It may be that non-cont. governs our corner of the universe, but is not valid over every domain, even of physical reality–but since it is valid in our own domain, and we evolved in a way that presupposes its validity, we are literally incapable of understanding, conceiving, or even really thinking about what things are like in those parts of the universe where non-cont. isn’t valid. “Incapable” might be too strong here–it might just be very difficult. This may be analogous to the way that, since we evolved in a domain where medium-sized gravitational physics are the most important governing principles, we find it very easy to think in terms of them*, and progressively more difficult to think about physics at other scales, involving other forces.

*Really, it’s not gravity that we find it easiest to think about, but a sort of false “gravity” that involves, not universal attraction, but a general tendency for objects to move “downward.”

15

Well, it would be an invocation of the anthropic principle to say that we live in a universe where the math is the way it is, because if the math were different, we wouldn’t be here. But I’m no great fan of such excuses (and I’m not even sure it would be correct, in this case, since it’s certainly not inconceivable that a universe based on a much different mathematical framework nevertheless could exhibit sufficiently similar phenomena).

16

Hmm, I’m sorry if I misread you. I thought your question to be asking about the properties of the universe – “Is this car blue?” --, rather than its foundations – “Must cars be blue?”. Even if the former question were answered in the affirmative, this may merely tell us something incidental to the existence of the universe, rather than something fundamental about its structure. However, it seems that we are actually thinking in very similar directions on this.

As for necessity – I think it is well possible that there might actually just be one (or a small number of) possible way to make universes, and that thus the universe is the way it is because it couldn’t have been any other way; that there is probably just one way (or a few ways) to mesh logical principles, mathematical models, and physical laws together to get one big, ‘working’ whole (I want to evade common terminology like ‘consistent’ or ‘coherent’ as they already depend on logical notions ultimately probably germane to our universe; but then again, anything I could possibly write or think does), that, for instance, if you didn’t have non-contradictoriness, you also wouldn’t have a universe.

(And regarding the ability to exhaustively search through the whole universe for contradictions, well, let’s take that more as illustrative, I recognize the problems inherent in the notion you mentioned. But there certainly are conceivable universes where you could do such a thing – if a universe only consists of things A, B, and C, and A does not contradict B does not contradict C and vice versa, that universe is free of contradictions, but we don’t know if that’s because it has to be, or just because it happens to be.)

17

Are there any branches of mathematics that aren’t used in some sense in any of the natural sciences, or used in any of the other formal sciences?

18

I agree. And I’m sure ITR Champ does it himself all the time. Almost anything rational humans do has to do with measuring happiness, fulfillment, and other such “higher” things . Is the happiness and utility I expect to gain from owning and wearing that pair of jeans worth $32.50 to me? Is a night out with this girl worth $40+(g)allons gas+(t)ime+a million other considerations?

I especially find fault with ITR’s assertion that

Just because measuring such things is difficult doesn’t mean they are inherently impossible to quantify. Everything can be measured, you just need the right ruler.

All of these things,

obviously occur to varying degrees and in different shades, (except personality, which is a straw man because nobody reasonably claims to be able to quantify “personality” as a whole numerically) and when you see a starving child picking at his empty rice bowl and another child laughing and running in the back yard with her dog, it’s not hard to compare their relative well being, happiness, despair, or whatever else numerically.

If you’ve ever paid for anything or spent time for any cause, you’ve measured a “higher” value in terms of dollars or minutes, both of which are mathematical representations.

19

There’s many different verbal ways of thinking about one of natures true oddities - a black hole.

As you fall toward the horizon you may consider yourself to be approaching c. But in reality it must really be rushing towards you at c (photons are trapped there and they must always travel at c.)

As you fall in you would observe virtual particle pairs being created, but if you’re suspended just outside then real particle pairs are being created.

To a faraway observer time comes to a stop at the EH, but if you’re falling in time passes normally. In effect there’s a discontinuity here but then again there really isn’t.

If you’re suspended just outside the hole you would be roasted alive by the blue shifted radiation from the rest of the universe, but on the other hand if you’re a little closer to the horizon, the light cone closes up and you’re not affected at all.

The math says one thing and one thing only.

20

It describes how things are.

It seems to me that it is a rule of language. The universe doesn’t have to follow it.

For instance, light seems to be both a wave and particle. Fuzzy logic is more appropriate for the universe because of quantum physics.

I guess I was actually addressing the notion that logic is bivalent, when it doesn’t have to be.

The liar’s paradox shows (IMO) that logic isn’t bivalent, and by extension that the law of non contradiction isn’t universal with all the formal systems. If you try to cut it up or change it, then you are missing the fundamental nature of the question.

“This sentence is a lie” is neither true or false.

You weren’t writing this to me, but I wanted to say that I don’t understand what you mean by ‘govern’ in this sense. Are you supposing that the LNC is a force that acts upon things?

I would say that the LNC isn’t a governing factor in our universe, as that doesn’t make a lot of sense. It seems to mean that the LNC is some how existent.

I don’t see how it’s amazing, since the universe isn’t reducible to bivalent logic. To me, this is like arguing that the rules of chess are amazingly consistent among chess players.

I guess I don’t get it.

This is where I feel you are getting off track. In what sense do you mean it governs anything?

I do not think they govern the universe at all. I think they are the foundations for our rational thought.

For instance, a car is a car and not not a car because we have defined a car to be something that suits the definition of a car. This is tautological, of course, but it’s how we define things.