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The Relation Between Mathematics and Physics by Paul Dirac

96 points rramadass 3 days ago 30 comments

rramadass 3 days ago | parent

Also see Relationship between Mathematics and Physics - https://en.wikipedia.org/wiki/Relationship_between_mathemati...

wolfi1 5 hours ago | parent

there is a quote from Pauli about Dirac : 'There is no God and Dirac is his prophet' Heisenberg and Dirac had different opinions on the existence of a god and Pauli, asked for his opinion, had the former to say

Tazerenix 4 hours ago | parent

There's a quote Atiyah has about spin geometry which he heard from his advisor Hodge, who had an office next to Dirac at Cambridge.

Only two people understand spinors: God, and Dirac. And Dirac's dead.

general_reveal 5 hours ago | parent

Haven’t even made too far down, but:

”There is no logical reason why the second method should be possible at all, but one has found in practice that it does work and meets with reasonable success. This must be ascribed to some mathematical quality in Nature, a quality which the casual observer of Nature would not suspect, but which nevertheless plays an important role in Nature's scheme.”

I mean, Hume answers this, but so does time, and I guess they both answer it together. Our observed measurements are true for the certain time period we’re in and remain true while we are in that window. To conclude with Hume, it’s all true until it isn’t, in time (well I suppose even in timelessness, entire properties can change - even so, with respect to measuring, I believe that requires a physical step function so we can’t avoid time here :)).

Aside:

”This is a quality which cannot be defined, any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating.”

Thorough worshipping of Math and “Mathers”, like as if they are a gift from God. Again, let’s stick with Hume, you’re the shit until you’re not. Einstein went to death not understanding Physics as he wanted to.

Hey, I’m a programmer, I didn’t necessarily want robots to do my job either. God loves that essay Hume wrote, truly.

It’s all fun and games until God mindfucks your understanding.

n4r9 4 hours ago | parent

This is probably a reference to Hume's problem of induction, i.e. that observing a regularity in the past cannot justify assuming that it will hold in the future. It's vaguely related to the article, but talking about "time windows" and "Mathers" muddles the point.

general_reveal 4 hours ago | parent

Is it vaguely related? He wondered why measuring anything even works. Why wouldn’t the measurement disappear, thus making all this measuring pointless? Because, apparently the measurement(s) are true each time-step, and it will be true for that time, until it’s not true. Why it reasonably works well for us? Well, God , I guess. Same way the Sun doesn’t burn our skin off (God, I guess, but some have issues with that).

Sorry, I had to critique that immediately because I could already see that other second paragraph I quoted coming, as in, this author probably worships Math, Science, and those that do it, certainly not God, because he was unable to characterize the phenomena of it as related to God.

n4r9 4 hours ago | parent

Yes, vaguely related. Dirac is not wondering why measurement itself works, but why mathematical reasoning applies so effectively to the measurement results.

I think it's valid to be surprised that all observations of a physical phenomena up until this point have obeyed a beautiful and (relatively) simple mathematical relationship, without even thinking about predicting the future.

GoblinSlayer 3 hours ago | parent

Measurement works because universe moves in a simple way. As Hume indicates, you need very complex motion for irregularity, maybe even sentient, but sentient universe is fairy tale.

otabdeveloper4 3 hours ago | parent

> Math, Science, ..., certainly not God

Pretty much everyone worships some sort of God. (The exceptions are non-self-aware people living moment-to-moment.)

What people argue about is the degree of personality and autonomy that God possesses.

lolakutty 4 hours ago | parent

>There is no logical reason why the second method should be possible at all,

One reason I can think of is if we consider that some regularity in the physical behavior is required for consciousness to evolve and so consciousness only forms in such worlds.

rramadass 3 hours ago | parent

Later down in the essay, Dirac himself answers this;

With this kind of cosmological picture one is led to suppose that there was a beginning of time, and that it is meaningless to inquire into what happened before then. One can get a rough idea of the geometrical relationships this involves by imagining the present to be the surface of a sphere, going into the past to be going in towards the centre of the sphere, and going into the future to be going outwards. There is then no limit to how far one may go into the future, but there is a limit to how far one can go into the past, corresponding to when one has reached the centre of the sphere. The beginning of time provides a natural origin from which to measure the time of any event. The result is usually called the epoch of that event. Thus the present epoch is 2 x 109 years [actually 2x10^9].

One further point in connection with the new cosmology is worthy of note. At the beginning of time the laws of Nature were probably very different from what they are now. Thus we should consider the laws of Nature as continually changing with the epoch, instead of as holding uniformly throughout space-time. This idea was first put forward by Milne, who worked it out on the assumptions that the universe at a given epoch is roughly everywhere uniform and spherically symmetrical. I find these assumptions not very satisfying, because the local departures from uniformity are so great and are of such essential importance for our world of life that it seems unlikely there should be a principle of uniformity overlying them. Further, as we already have the laws of Nature depending on the epoch, we should expect them also to depend on position in space, in order to preserve the beautiful idea of the theory of relativity there is fundamental similarity between space and time. This goes more drastically against Milne's assumptions than a mere lack of uniformity in the distribution of matter.

Show respect to Mathematicians/Physicists/etc. many of whom had studied Philosophy and i dare say had a better understanding/insight of it than mere academic Philosophers. Why? Because they don't go off into la-la land but try very hard to have both observed evidence and a mathematical model of it match cleanly via a Theory.

For your edification, read first "Meditations on First Philosophy by Rene Descartes" followed by "Physics and Philosophy by Werner Heisenberg".

VorpalWay 5 hours ago | parent

Shouldn't this have a "(1939)" suffix on the title according to HN rules? ;)

sigmoid10 4 hours ago | parent

Should it though? I don't see such a rule.

https://news.ycombinator.com/newsguidelines.html

baxtr 3 hours ago | parent

unwritten rules

delichon 1 hour ago | parent

  Kaffee: Corporal, would you turn to the page in this book that says where the mess hall is, please?
  Barnes: Lieutenant Kaffee, that’s not in the book, sir.
  Kaffee: You mean to say in all your time at Gitmo you’ve never had a meal?

ape4 1 hour ago | parent

If an article is a few years old then it could be out-of-date. But if an article is almost 100 years old then its classic.

GoblinSlayer 4 hours ago | parent

>but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen

This is quantifiable, right? Should be possible to measure how many mathematicians favor ideas like turbulence, macroscopic quantum physics, anthropic principle, analog geometry, finitism. I think, mathematics doesn't really try to match physics yet.

huurtehoog 1 hour ago | parent

I think this passage is more about the fact that mathematics is a form of symbolic computation people create and it is weirdly congruent to the physical measurements and models of reality. Not that mathematicians see certain models and objects used in physical models in a favorable way. What favor even means here?

It is quantifiable: several mathematical objects and categories people invented to do symbolic computation in letters and published papers are somehow very useful to model and understand data from physical measurements of reality.

I highly recommend reading the "Unreasonable effectiveness of mathematics" essay by Wigner:

https://www.hep.upenn.edu/~johnda/Papers/wignerUnreasonableE...

DonaldFisk 54 minutes ago | parent

Dirac was talking about the fundamental laws. In favour of his point, and not known at the time he gave the talk, the standard model of particle physics is based on the symmetry groups U(1), SU(2), and SU(3), so Lie groups (which mathematicians find interesting) appear to have been chosen by nature.

empath75 6 minutes ago | parent

I sort of have this suspicion that the kinds of phenomena that we can even apprehend are pretty closely related to what we can reason about mathematically, and the reason that math seems to work so well to model behavior that we understand says more about how our brain works than how the universe works.

If a phenomenon isn’t model-able by relatively simple math it doesn’t even look like a phenomenon to us, it looks like a chaotic mess.

Which is to say that unless something has mathematical coherence it is more or less invisible to our experience.

taaha47 4 hours ago | parent

"I would like to put forward a suggestion as to how such a scheme might be realized. If we express the present epoch, 2 x 109 years, in terms of a unit of time defined by the atomic constants, we get a number of the order 1039, which characterizes the present in an absolute sense. Might it not be that all present events correspond to properties of this large number, and, more generally, that the whole history of the universe corresponds to properties of the whole sequence of natural numbers? At first sight it would seem that the universe is far too complex for such a correspondence to be possible. But I think this objection cannot be maintained, since a number of the order 1039 is excessively complicated, just because it is so enormous. We have a brief way of writing it down, but this should not blind us to the fact that it must have excessivly complicated properties."

this cannot be an early hint to the result made by chaitin; Chaitin’s Incompleteness Theorem?

rramadass 3 hours ago | parent

Related;

From Heisenberg to Goedel via Chaitin - https://arxiv.org/abs/quant-ph/0402197

rramadass 3 hours ago | parent

Just to clarify; The term "2 x 109" is actually "2 x 10^9" or 2 billion years which was considered the present epoch i.e. accepted age of the universe at that time.

Dirac large numbers hypothesis - https://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis

DonaldFisk 2 hours ago | parent

Correct (but now we know it should be somewhat higher than that), and 1039 should be 10^39, one of Dirac's Large Numbers, the size of the observable universe in atomic units, and also the ratio of electromagnetic to gravitational forces in the hydrogen atom.

rramadass 2 hours ago | parent

Right. Added the wikipedia link to Dirac's LNH above.

Also found this; The fundamental constants and their variation: observational status and theoretical motivations - https://arxiv.org/abs/hep-ph/0205340

Xmd5a 3 hours ago | parent

> The dominating idea in this application of mathematics to physics is that the equations representing the laws of motion should be of a simple form. The whole success of the scheme is due to the fact that equations of simple form do seem to work. The physicist is thus provided with a principle of simplicity, which he can use as an instrument of research.

https://arxiv.org/abs/math-ph/0009007

> Occam's Razor as a Formal Basis for a Physical Theory

> We introduce the principle of Occam's Razor in a form which can be used as a basis for economical formulations of physics. This allows us to explain the general structure of the Lagrangian for a composite physical system, as well as some other artificial postulates behind the variational formulations of physical laws. As an example, we derive Hamilton's principle of stationary action together with the Lagrangians for the cases of Newtonian mechanics, relativistic mechanics and a relativistic particle in an external gravitational field.

anthk 3 hours ago | parent

Now dive down into Math with Penrose and David Bohm with the implicate order.

rramadass 1 hour ago | parent

Quite independently of the Mathematics, Bohm's ideas of "Implicate and Explicate Order" are very interesting and comprehensible by the layman at a high-level - https://en.wikipedia.org/wiki/Implicate_and_explicate_order

In particular, students of Hindu Philosophical Schools will find lots of parallels here. Bohm was heavily influenced by Jiddu Krishnamurti which inspired his take on quantum theory.

Bohm also wrote a great book, "Quantum Theory" based on the Copenhagen Interpretation which contains separate parts on "Physical Formulation" and "Mathematical Formulation" of quantum theory.

3lambda 20 minutes ago | parent

Given its strength in mathematics, it seems likely that AI should be of great help with discovering new physics. We just need to teach it to ask interesting questions.

jbmchuck 7 minutes ago | parent

How does this comment relate to the article?