A naturalist account of the limited, and hence reasonable, effectiveness of mathematics in physics by Lee Smolin

Your essay is extremely interesting, as usual. I have read and appreciated it as I appreciated your books and papers. Your concept of time is enlightening. However, I would just like to note that it is not just a matter of Naturalists vs Platonists. There are also people thinking at mathematics as a language to speak about the Nature (Galilei, Bohr, ...).

Best regards and good luck for the competition!

LF

I found your essay easy and enjoyable to read. Getting to the end I am a little puzzled because the conclusion seems to almost contradict the opening and abstract "My aim in this essay is to propose a conception of mathematics that is fully consonant with naturalism. By that I mean the hypothesis that everything that exists is part of the natural world, which makes up a unitary whole." Lee Smolin

Some thoughts you may or may not wish to ponder:

a, If all that exists is part of the unitary whole, not evoked, should we, in your opinion, then consider mathematical objects do not exist?

b, You say they "do not exist before being evoked by human imagination" What about balls and paperclip magnetic pyramids and lines on paper. Aren't these as much mathematical objects as the ones imagined? Is it their vulnerability to change that excludes them? As you later say the properties of mathematical objects are independent of time.

c, What came first the real substantial object which is idealized by imagination or the imagined object then identified with real objects? Perhaps that is a question for neurosciences.

You give the mathematical objects the property of being independent of time, and having to be evoked by the mind so- don't they then exist in an imagined realm. Thus Not in contradiction with the Platonic view, according to which, mathematical truths are facts about mathematical objects which exist in a separate, timeless realm of reality, which exists apart from and in addition to physical reality.

Or -and now it gets very interesting, in my opinion, will you put the minds full of imagined mathematical objects within the not evoked real universe and so have the evoked as an internal subset that is yet in some way apart from other subsets of the unevoked universe?

Good luck and kind regards Georgina

Dear Lee,

In Boethius's Consolation of Philosophy, the Muses of Philosophy chase away the Muses of Poetry by calling them bad names ("theatrical whores", etc.). One way to read this is to emphasize the difference between lamentation (Poetry) and consolation (Philosophy). Another way is to note that the Muses of Poetry dictate (evoke) to the suffering Boethius truths that are only temporal and encourage him to come to terms with his present condition. The Muses of Philosophy, on the contrary, tell eternal, atemporal truths and promise to heal Boethius. If we are the unfortunate ones who suffer from inadequate physical theory, then shouldn't we place our best hope in consolation by the second kind of Muses? Isn't it a matter of greater hope that a better physical theory will be possible if we hear it in a dictation that is underwritten by the rational timeless argument?

Best wishes,

Alexei

Dear Prof. Smolin,

I totally agree with you and cited your works in my writing.

Best regards,

Leo KoGuan

Dr. Smolin:

Nice essay on a topic that I suspect will evolve immediately in people's minds.

It would seem that evoking (or evolving) requires some level of (time-bound historical) continuity to the process. Even if someone were to present a consistent 25-dimensional theory today, it is unlikely that it would be accepted in the world (today) - simply because the connections between that theory and our current ones would not be traceable.

The question of what has been evokes comes up, as does the question of the loss of such an idea if it is rejected today.

Beyond that, what we might want to look deeper into is the evolving character of mathematics. If the greeks did not have an adequate conception of number for today's science, why do we think we have an adequate conception for tomorrow's science?

While the answer might be 'we are locked into today's mathematics', I agree with your comment: "... that the goal of philosophical argument is not to arrive at a logically perfect position but to suggest novel hypotheses for science to examine and develop."

Thank you, Donald

Dear Lee,

I very much enjoyed your essay and personally appreciated your naturalist stance. I try very hard not to judge the essays so much as to whether I agree with every point, but rather whether the author made a clear, cogent, and insightful case focused on the topic at hand. With that in mind, I believe that your essay played the role of an effective exorcism on the forces of mysticism in physics. I agree with Edwin Klingman above who wrote "I do not believe I have ever seen such a devastating analysis of the sheer uselessness of the Platonic idea of a mathematical realm outside of space and time."

Thank you!

I also appreciated the distinction you made between wonder and mysticism. I believe that a good number of scientists in foundations harbor a secret (or not so secret) love or appreciation of the apparent or perceived mysticism in physics. While many speak as if they believe in an underlying simplicity, there is often a gravitation toward more mystical ideas and sophisticated mathematics. To combat complexity and excise mysticism, in my essay I focused on the practical problem of additivity and the insights it provides into mathematics and physics. It is refreshing to have read your essay which takes a more philosophical approach.

Hi Lee,

I am in general agreement with about half of your essay. I entirely agree that we will never be able to completely eliminate experimentation. In fact, it should be at the core of everything we do. In that sense, mathematics is merely a description.

That said, I am troubled by the broader claims you laid out about physical laws being "evoked" and "timeless." It seems to me that this would be particularly problematic for observational cosmology which is quite literally looking into the past to infer information. Are you seriously saying that cosmological laws don't exist until human beings "evoke" them? And why would human beings be particularly special, in that regard? Many animals species can do simple math. So it strikes me as being a tad solipsistic.

In addition, the assumption that the concept of time is universal and somehow immutable seems to contradict the fact that it is entirely meaningless to certain systems (e.g. a single, free electron). It seems to me (as well as to some others) that time is an emergent phenomenon.

Finally, a minor point: I would argue that logic is not a fourth, separate concept in mathematics. Rather I would argue that it is more fundamental than mathematics in general. All of mathematics is built on logic.

Cheers,

Ian

Dear Lee,

I think there exists yet another aspect of the naturalism to consider. This is the evolution of geometrical structures. These structures that are not abstract mathematical platonic objects but real pieces of dynamical elastic medium - spacetime. These pieces evolve. The language of mathematics (like 3+1 manifold) we use is obviously only our tool (the baggage we have invented) to communicate and develop the description of that real medium . That view slightly differs from yours as it allows to explain what the reality is (a dynamical evolving elastic spacetime) and not only the way it works. Knowing many of your publications I get a strong impression that this is what you really expect from physics. Sorry, maybe I am wrong.

I argue that we are able to make general predictions directly from a set of geometries (described by the mathematical language). An example is the set of Thurston geometries. From the proof of the geometrization conjecture (by Perelman) and correspondence rule we can be convinced that another geometries (than these 8) cannot exist not only for mathematicians but also in reality. In my opinion two core concepts are used here. Both you have mentioned - geometry (Thurston geometries) and logic (Perelman proof).

You claim that "Nature has ... the capacity to create kinds of events, or processes or forms which have no prior precedent." This does not mean that Nature does not sticks to some rules e.g. the rule of evolution that only steady entities can exist. And it does not mean that these rules are not eternal. We cannot know that as we are not eternal. For sure we cannot predict the future of the universe. This is the feature of evolution. It does not mean that we cannot discover general, timeless rules of evolution.

I would appreciate your comments http://fqxi.org/community/forum/topic/2452

Thank you for the essay and for many great publications.

Jacek

Dear Mr. Smolin,

I appreciate your reliance upon common sense; a conception that is not as common as commonly believed. I also concur with your view that there is "no perfect correspondence between nature and mathematics."

You state that "it is essential to regard time as an essential aspect of nature". While I endorse this statement as being correct, I find it difficult to imagine how time or nature could be represented otherwise.

As the prefix "Uni" clearly implies, there is only one all-there-is. We agree on the existence of a singular universe which embraces everything that exists, whether we have acquaintance with such "things" or not. Your statement that "There are no eternal laws" also rings true insofar as laws are too rigid to accommodate eternal changes in nature's performance. The concept of "principles" better describes the causes or natural biases that effect change.

While, as you state (with regard to biology and mathematics) that in a timeless platonic world "there is a potential infinity of FAS's (formal axiomatic systems)", such potentials are merely infinite opportunities to exist, not existences per se. In the natural world the notion of timelessness evokes senselessness!

I suspect that in editing your essay you overlooked the need to delete the statement "How can something exist now but also exist timelessly" - though you did omit the question mark (?) пЃЉ.

Again, "How can something exist and not be made of matter?" is questionable. Time, space, energy, relations, the intellect, volition and the affections are not made of matter. No matter!

The statement "mathematical arguments are just finely disciplined cases of the usual rational thinking that all humans constantly engage in to understand their world" seals the fate of your naturalist position in your favour, i.e. that mathematics is reasonably effective in physics.

In reducing nature to its ultimate quality, all that exists are relations. That is exactly what mathematics and physics deal with and why they are complementary visions of the same world.

Congratulations, and thanks for the trip.

Gary Hansen

I think that Smolin's naturalistic perspective is the only reasonable position. He doesn't have to make Platonism into a straw dummy to defend it. His idea of evoked properties is something new, and a significant contribution to the philosophy of science (my field). The only negative criticism I have is that he focuses on topology as the basic connection between math and physics. There are other basic connections that cannot be reduced to topology

Dear Lee Smolin,

You take some pretty hard hitting comments about mathematics ie. "Mathematics thus has no prophetic role in physics, which would allow us an end run around the hard slog of hypothesizing physical principles and theories and testing their consequences against experiment.". I am not complaining at all, in fact a bit of a contrarian view forces a person to give some real thought to ideas they may have taken for granted.

You present a lot of solid arguments to support your ideas, as in the "small correction terms" that have to be made in may calculations, pointing out "This fact of under-determination is a real problem for those views which assert that nature is mathematical or that there is a mathematical object which is an exact mirror of nature".

I enjoyed reading your essay. I am not sure I am fully convinced at the end, but appreciate a different perspective to ponder.

Regards and best of luck in the contest.

Ed Unverricht

You argue that there is only one world in naturalism, which makes up a unitary whole. I agree with a singular universe, but it is also true that there are things in that natural world that are unknowable and therefore not revealed by reason alone. So there must be at least two portions to your universe; belief and reason.

And of course time. Time is as you say a succession of moments, but time is also a decay of those moments and it takes both dimensions to tell time with a clock. So the natural world must have time as well as matter and action to be complete.

1.0, entertaining

1.5, well written

1.8, understandable

1.5, relevance to theme

5.8 total

Lee,

This is a terrific essay, and I think it pays to reread it several times. Some of the many themes that interested me are:

1. The truly new: both nature and human beings can, and do, invent the new: "Nature has within it the capacity to create kinds of events, or processes or forms which have no prior precedent. We human beings can partake of this ability by the evokation of novel games and mathematical systems."

2. Emergence and evolution: in novel games and nature, the novelty "gives a precise meaning to the concept of emergence" and evolution. But "In a timeless world emergence is always at best an approximate and inessential description because one can always descend to the timeless fundamental level of description".

3. Platonic realm: belief in a platonic realm can "add nothing and explain nothing" and must "involve us in a pile of questions that...cannot be answered by rational argument from public evidence."

Cheers,

Lorraine

Dear Lee,

Thank you for the most incisive formulation and defense of naturalism I have seen! One thing that I especially like about your stance is the implicit use of Occam's razor to make room for possibility without questionable claims for necessity!

I fondly recall a dramatic event in a QED class I had with Feynman in which he railed against Axiomatic Quantum Mechanics, declaring " If anyone tells me to every observable there corresponds an operator such that . . . (continuing to recite an axiomatic mantra) . . . If anyone tells me that, I will defeat him! I will CUT HIS FEET OFF!!" -- dramatized with a grand cutting gesture across the ankles.

Respectfully.......David Hestenes

Dear Lee,

You have mentioned "By that I mean the hypothesis that everything that exists is part of the

natural world, which makes up a unitary whole. This is in contradiction with the Platonic

view of mathematics held by many physicists and mathematicians according to which, mathematical truths are facts about mathematical objects which exist in a separate, timeless realm of reality, which exists apart from and in addition to physical reality."

So a new conception of mathematics is needed which is entirely naturalist and regards mathematical truths as truths about nature. In this essay I sketch a proposal for such a

view. The key it turns out is the conception of time. I propose that to get a conception of mathematics within naturalism it is essential to regard time as an essential aspect of nature, in a sense to be specified shortly. I thus propose to call this new view, temporal

naturalism.

1. The singular universe: All that exists is part of a single, causally connected universe.

The universe and its history have no copies, and are not part of any ensemble.There is no other mode of existence, in particular neither a Platonic realm ofmathematical objects nor an ensemble of possible worlds exist apart from the single

universe.

2. The inclusive reality of time: All that is real or true is such within a moment, which is one of a succession of moments. The activity of time is a process by which novelevents are generated out of a presently existing, thick set of present events. Thereare no eternal laws; laws are subsidiary to time and to a fundamental activity ofcausation and may evolve. There is an objective distinction between past, present and future.

This is what I have emphasized in my Mathematical Structure Hypothesis that mathematical and physical reality don't exist independently and both are creation of Eternal Vibration which creates the entire Universe. thats why mathematical structures and physical reality both are effective to solve each other. This further sorts out your concern that "If we give up the idea that there is a mathematical object existing in a timeless Platonic realm which is isomorphic to the history of the universe, we still have to explain why mathematics is so effective in physics."

The paradoxes, inconsistency,contradiction exist because physical reality has been equipped with time and frame dimension but mathematical reality are taken into timeless, frameless dimension. But as my MSH states that both originate from Eternal Vibration, mathematical reality should also be extended in time/frame dimension like physical reality and made dynamic. Skolem had also shown that axiomatization of set theory leads to relativity.

In the Absolute there are no space,time & causation but we allow them in physical reality and take time into timeless space without causation. This fundamental discrepancy must be removed so that mathematics could evolve beyond paradoxes,contradictions dynamically.

Anyway you have written great essay.

Regards,

Pankaj Mani

Dear Dr Smolin,

A good essay. The natural question then is; Do you feel it's possible that a 'simple' logical answer to our very incomplete understanding may be hiding right before our eyes?

The more important question that leads to is then; How would we recognize it?

I tried a different method and, however unlikely, found a mechanism that self evidently works. Would you agree electrons (or e+/-) re-emit photons at c in their local centre of mass rest frame? The implications are important. As it seems you may be one of the very few able to 'see' the implications (I have your books) I hope you might be able invest a few minutes to look. My last 4 essays (all finalists) presented glimpses, perhaps better this than this years is; 'The Intelligent Bit'.

I consider your essay excellent and worth a top score, though the matter of yet being exactly on the right trail may be a different one. I hope perhaps after the above you may be interested enough to read mine. It seems the (early toy) model will remain 'invisible' otherwise.

Many thanks if you can make the time.

Peter Jackson

Dear Lee Smolin,

Please forgive me for my remarks. Let us regard a Platonic view related to the Cave story.

Then perhaps change the fire behind the people in the Cave with lights of different colors. When the green light is switched on, we the prisoners say.... look, a mathematical description of physics. Note we look at our shadow protected on the wall in front of us. Then when the red light is switched on and projects from a different angle and position, we say... look a physical picture. Still we are looking at a projection of ourselves on the wall in front of us.

So, a Platonic view connects the mathematical with the physical by noting that in both cases we are dealing with a projection of ourselves.

To "projection of ourselves" in a previous post, I would like to add that this is intended as the connection between the physical and mathematical platonic idea.

Of course, "that what is projecting" i.e. the "lights" behind the prisoners have their own characteristics. So the notion that "science is not possible" in such a conception, is a mistake.

When we ask a fundamental question like the relation between math and phys reality, it is perhaps necessary to introduce the the "thinker" / " observer" too and his/her characteristics and limitations. Those limitations are unknown until we find them. Wave-particle duality could very well be based on such a limitation/characteristic.

There are no eternal laws; laws are subsidiary to time and to a fundamental activity of causation and may evolve.

In my essay I present a mechanism which could give a rise for time, hence I claim that your principle might be "wrong".