The limited, and hence reasonable, effectiveness of mathematics in physics
arxiv.org
arxiv.org
It could be as well argued that there is only one core concept: computation. (Or maybe "symbol".)
The interesting question is: of all the myriad possible things that are computable, how is it that we are able find the unfathomably small subset whose behavior corresponds to (what we perceive as) physical reality? That was the question Wigner was asking, and I don't think Smolin answers it at all.
The fact that Smolin thinks these are distinct things shows just how out of touch he is, and is ironic given his hatred of mathematics.
Wait, what? Are you going off of a particular polemic of his?
Because it's hard to imagine someone who had to have learned enough math to do string theory, to be a true math hater, even if (I am assuming) he rants.
Isn't that simply because of the Anthropic principle [1]? We are observing this computation simply because we are part of it. Other creatures may be observing different computations (but we will never be able to ask them because they live in a different reality).
(1) Anthropically, you should not be "surprised" to observe a world in which 100 expert marksmen all missed from point-blank range, because the very fact of your being able to observe anything at that point makes that world necessary.
(2) But you nonetheless should be "surprised" that such a low-probability event has occurred, and you would be justified in wondering whether some as-yet-unaccounted-for factor meant that the true probability was not what you would have estimated it to be.
Anthropic arguments, in this view, slyly try to switch between different meanings of "surprised" (or a variant, depending on the level of formality and the particular terminology chosen) to make it look as if (1) actually answers (2) when it doesn't.
If you are a Christian, then you most likely believe in a single objective universe, which is the one we live in (I'm not a Christian so I won't go as far as saying that a Christian must believe this, although many theologians do).
If you are not a Christian, you might have more flexible beliefs that allow for the anthropic principal.
My thesis (12 years ago, before I learned that coding paid a lot better than philosophy) was a survey of how the argument from design had developed post-Darwin, and part of that involved digging into arguments being made by physicists who had no particular religious affiliation but still felt that the Goldilocks-ness of our universe was something that needed explaining.
And it turns out a lot of those people find anthropic arguments to be, at best, unsatisfying; at worst, they can point to logical fallacies underlying anthropic attempts to explain away the conditions of the universe we find ourselves in.
The interesting arguments, to me, fell into two camps:
1. Natural selection operating on the level of entire universes. Several different theories postulate singularities as a source of new universes, which in turn means that a universe in which conditions allow for significant gravitationally-bound accumulations of matter will have more "offspring" than a universe in which they don't. Thus, physical constants which are conducive to life are likely to be more common than otherwise expected, simply because (coincidentally) those conditions are the ones which give rise to significant gravitationally-bound accumulations of matter.
2. Arguments in favor of simulated universes. Paul Davies (if you want a name to look up) has a few layman-accessible articles arguing about the probability of this, and working from the fact you only need one life-supporting universe -- which is likely given various of the multi-universe theories -- to kickstart the process and wind up with a far larger number of life-supporting simulated ones.
There are also people who make arguments in favor of a more traditional religious conception of "God", but do so from the standpoint of regularities in physical law. Those are less interesting to me personally for a few reasons, but they are out there.
So if you get killed in that firing squad, it is more like a few brain-cells of a much larger creature (humanity?) got killed.
Or am I missing something?
There are myriad branches of mathematics; he's emphasizing these these four as fundamental.
The answer is: we are humans, doing human things within the scope of human capabilities. If there are ideas that are inexpressible by us, we can't possibly know about them. If there is physics profoundly beyond our ken (what lies behind the quantum veil, for example) we simply don't know about it.
Mathematics is a natural language (as physicists use it) to describe nature to ourselves. The fact of the knowing subject, and the activity of the knowing subject, cannot be left out without leaving a central mystery, which always amounts to "Why does the knowing subject do what they do?" (like restrict math to Smolin's four key categories of number, geometry, algebra and logic). If you imbue some mystical subject-free "mathematics" with these properties, rather than the activity of the knowing subject with them, they will remain mysterious.
This also explains why mathematics is so very, very bad at describing reality: http://www.tjradcliffe.com/?p=381
That's not an equivalent question. Any repertoire of N distinguishable symbols for N>1 is essentially equivalent.
But there's no reason a priori to believe that the laws of physics should be modellable with mathematics at all, let alone that we should be able to figure out what those mathematics are, let alone that they should turn out to be simple enough that the model (or at least a very significant chunk of it) can fit in a single human brain.
Consider dreaming: when you are in a dream state you are living in essentially a solipsistic world where science doesn't work. There's no inherent reason why that could not be totality of your existence. It's just an accident of biology that you wake up occasionally and get to experience the "real" world, which we consider "real" because it seems to behave according to mathematical laws. The existence of such experience is not a given.
"As Roberto Mangabeira Unger and I argue in our new book The Singular Universe, the most important discovery cosmologists have made is that the universe has a history. We argue this has to be extended to the laws themselves. Biology became science when the question switched from listing the species to the dynamical question of how species evolve. Fundamental physics and cosmology have to transform themselves from a search for timeless laws and symmetries to the investigation of hypotheses about how laws evolve."
and
"Neurosciences are a fabulous area to work in, ripe for great discoveries. I’ve always felt this and indeed the only alternative to a career in physics that ever attracted me was a brief flirtation in college with neuroscience. But that is a field which is as bedeviled by outdated metaphysical baggage as physics is. In particular, the antiquated idea that any physical system that responds to and processes information is isomorphic to a digital programmable computer is holding back progress."
[1] http://www.cornell.edu/video/nima-arkani-hamed-spacetime-is-...
Yes, this has gained increasing acceptance ever since its first (?) suggestion from the late illustrious John Archibald Wheeler in the 1960s, and it is a very compelling idea, although yes, it makes it hard to conceive what we may be talking about.
> like Nima Arkani-Hamed they seem to disagree,
I missed it, who's disagreeing with whom about what? The parent comment doesn't seem to have content disagreeing, unless I'm missing something.
Aka if the rule is light speed = age of universe then that's just the rule not speed of light is X and slowly changing.
This is a special case of a Noetherian invariant: all global conservation laws follow from a symmetry.
https://en.wikipedia.org/wiki/Noether%27s_theorem
> Indeed, properties of spacetime change from one location to another, so one could be inclined in formulating only "local laws of physics", and stating those laws change from location to location according to mass distribution among other things.
Ummm...you got it right on the first try, whereas this restatement has gone into the weeds. Local conditions (values of variables, you could say) change; the laws do not.
The final part seems to go back on track again, but by then I'm confused what you're saying.
Perhaps you're just using some kind of counterfactual as a rhetorical method or something.
> This is a special case of a Noetherian invariant: all global conservation laws follow from a symmetry.
No, it isn't. I'm afraid you don't know what you're talking about.
Spatial translation symmetry lead to conservation momentum, time invariance lead to conservation of energy. No symmetry leads to "physical laws are same everywhere". That's just something you have to assume and can't derive.
"Spatial translation symmetry" is what I usually think people mean when they say "physical laws are same everywhere" -- nor is that foolish of me, that's exactly what it sounds like.
> I'm afraid you don't know what you're talking about.
Quoting from the link I posted,
> Noether's (first)[1] theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law.
What I said was a simplified rephrasing of that, for someone who apparently did not understand physics.
Jeez. If I even so much as know the term "Noetherian invariant", then I am not totally clueless, even if you disagree about my phrasings.
Get over yourself.
Wait, you are pissed that I criticized this phrase of yours:
> Indeed, properties of spacetime change from one location to another, so one could be inclined in formulating only "local laws of physics", and stating those laws change from location to location according to mass distribution among other things.
That phrasing is either someone who doesn't know physics, or, since you claim to be a theoretical physicist, alternatively it means you are a terrible writer.
Neither is a reason to get rude with me.
The thing is, it's not meant to be an insult --you literally don't know what you're talking about (or you're trolling). When we talk about symmetry, we mean a symmetry of the Lagrangian (density) that describes the system. The Lagrangians everybody write down a priori assume that laws of physics are same everywhere.
The spatial symmetry for example just means L(x, v, t) = L(x', v, t) where x and x' are different by a small amount. An example of this in the classical context is a flat space-time with no potential (U = 0) everywhere. This just means the space "feels" the same everywhere (that is, there are no forces), which in turn means momentum will be conserved (due to the form of the action). This doesn't mean we have proved the laws of physics are same everywhere. That's an assumption we had from the beginning when we wrote down the Lagrangian.
Lack of spatial invariance in realistic systems doesn't also mean "laws of physics depend on where you are". Again, we always assume that it is true; the broken spatial symmetry just means there is a non-uniform field.
And the term that you proudly "know" ("Noetherian invariant") doesn't exist, so I wouldn't hurry and say "you're not clueless" about what you're talking about. You might wanna read that Wikipedia link you gave.
Further, my past experience is that physicists tend to frequently misunderstand physics that is outside of their own narrow speciality, so since you're more interested in being insulting than in educating, I can assume nothing about your supposed superior understanding. Seems like just ego.
I was just pointing out to a piece of misinformation before someone takes it to be true. And again, it wasn't meant to be an insult, I literally meant it and it is true --you literally didn't know what you were talking about. I only meant to state a fact. There's no need to make this something personal. For what it's worth, I'm sorry if it offended you.
In my view, not knowing things is not something bad. You can always learn things. But spreading misinformation about something you don't know (and insisting on it) is not.
> Further, my past experience is that physicists tend to frequently misunderstand physics that is outside of their own narrow speciality
??? Are you talking about physicists (faculty), or students? Physics is a lot more connected than you think, and I haven't yet met a single physicists who doesn't know undergrad-level classical mechanics.
Edit: After reading your reply, I take that "sorry" back. For me, you're not someone worth spending even single second for. You can take this as an insult if you think otherwise.
You thus contradict yourself for purely rhetorical reasons.
Worse yet, saying "you are wrong" is not correcting misinformation for the sake of onlookers. You need to tell them why -- which you failed to do.
I don't think you interpret English very well, and certainly you don't explain well, so it's impossible to see whether this is an area of physics you actually know or not.
So far, I am doubting that you understand this area well enough to correct others. All I see so far are two beside-the-point nitpicks.
Explain physics or stop kvetching. If it's not your job to explain, then it's unhelpful to say "you're wrong but it's not my job to explain!" Fish or cut bait.
Okay, but when were the variations of chess invented? For example, instead of playing on an 8x8 board, you could play on an x*y board for any natural numbers x and y. Do we need to set x and y to specific values for that variation to "come into existence". Does it make sense for us to say that a game exists when it's never been played?
How specific do we need to be about imagining a game? If we just say that chess can vary in many possible ways by changing any of its rules, does checkers come into existence? Or suppose we write a computer program to generate variations on board games. Do I have to run the program for these games to "come into existence", or just write it, or just think about writing it? If the program has a bug so that it won't generate a particular board game, does that game come into existence when I document the bug, fix the bug, or verify the fix?
I don't think any philosophy is all that good at distinguishing between explored and unexplored concepts in the general case. But that's a problem mostly when you try to do philosophy. If you have a practical purpose in mind, it's much easier to come up with a useful working definition for the particular domain you're interested in.
We can objectively say that 1492 was an important year of first contact and we should talk about what happened. But it wasn't "discovering the New World" from the Native Americans' point of view.
Both in science and math, publishing a paper can be an important event and it's a physical process. But the first math paper published may not correspond to the first time someone had an idea or made an observation. If we're going to talk about a concept as a thing that exists in time and space, we need to take into account that ideas may be created, shared, and forgotten many times.
Perhaps we can think of Platoism as a way to avoid getting bogged down in the history of an idea by declaring it irrelevant for our purposes.
A lot of brilliant famous mathematicians in history have been befuddled on that point as well.
I think it's a linguistic issue, which perhaps you are hinting at.
If I say "that rock exists", in the real world, that meaning of "exists" is parallel to, yet different than if I say "prime numbers exist".
It's hardly the first time that one word meant more than one thing.
The linguistic "exists" is a "deictic" -- I can point to the thing.
The mathematical "exists" means "come, let us reason together, and I can demonstrate to you abstractly that this one thing is inescapably implied by this starting point".
(Phrasing borrowed from Leibniz's commentary on Calculus Ratiocinator)
They both have a similar final state, I suppose: that we are convinced of the "existence" of a thing, but nonetheless the existence in each case is in a separate realm.
http://www.amazon.com/Where-Mathematics-Come-From-Embodied/d...
Absolutely fascinating book.
One reason I find it hard to reject Platonism entirely is that the main alternative in the philosophy of mathematics, Formalism, looks to me like a kind of Platonism. This is because formalism replaces belief in the independent existence of mathematical objects (real numbers, integers) with belief in the independent existence of meta-mathematical objects (theorems, proofs) which most formalists conceive of in mathematical terms.
Ultimately I think that there is no a priori limit that we can place on how science is going to pan out, and how this will affect our philosophy. E.g. I find the idea of multiple universes distasteful and nonsensical (e.g. If everything possible happens, then how can we interpret probabilities? Why do we live in a universe where macro observations are consistent with applying the law of large numbers to quantum randomness? I can't think of any anthropic/observer based principle that would enforce this) but I can't yet rule out that that's really how things are.
How does that work with relativity, which says that the concept of simultaneity does not even exist in the sense that we understand it? [1]
BTW, if you also want to talk about the present, you can only talk about it at a particular location. If you want, you can define 3 "thin" boundary regions: the surface of the future light cone (boundary between absolute future & absolute elsewhere), the surface of the past light cone (similar), and the "here & now" (the single point in spacetime where all 3 regions meet).
Please excuse my quibble, but more precisely it is a classical/Newtonian concept rather than a relativistic concept.
"Human concept" might map to "folk physics" (the version of physics assumed by "the common man" humans from current hunter-gatherer societies and older societies etc.)
There are illustrations from the middle ages and from ancient Greece, if I'm not mistaken, that show the flight of an arrow to be a straight line up to apex, and then linearly down to target (rather than a realistic parabola) -- and although I've always wondered if an actual bow expert thought that's how it worked (unlikely), but apparently the non-archer philosophers of the day apparently did.
But I digress. You are clearly just pointing out the modern understanding.
Mathematics/logics is one of them. It says that the world is guided by principles. "Objects contain different amount of phlogiston and when they burn it is released."
Anthropomorphism is the other. I.e. there is one, or many human-like beings with personalities and feelings. And things that happen happen because the beings want them to. "If you offend the fire spirits your house may burn down"
Are there more?
Some people believe in karma. Maybe this is covered as a principle, however, even though it isn't scientific.
Perhaps solipsism might also count as an "explanation" of the world. I exist, so that explains the world.
Fate is also an explanation for what happens.