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

Hi Lee,

I'm puzzled about the status you ascribe to the fundamental principle enunciated here. It seems as if your basic idea is to try to figure out the consequences of accepting a "strong form of Einstein's principle of no unreciprocated action according to which there can be no entity A which plays a role in explaining an event B, that cannot itself be influenced by prior physical events." There are various questions on could raise about such a strong principle: logical (does it apply to itself, so it can't remain the same?); metaphysical (why should it have such a foundational status?) and epistemological (why think it is true?). Let's leave aside the first two and focus on the last. There are three possible positions here. The Heraclitean position is that everything real is in flux, which seems to be your position. The Parmenidean position is that nothing real ever changes. And the mixed position is that some real things change and some don't. We can reject the Parmenidean position since the physical world is real and changes. Newton, Einstein, etc. all hold the mixed view, and you are trying to hold the radical Hericlitean view, it seems.

Now let's grant that the view can be made logically consistent (which is not clear). The mixed view has no such problem of self-application: according to the mixed view, the truth of the mixed view itself can be one of the things that never changes. So unless you have some mystical source of knowledge, you can't know that the mixed view is wrong. And there is powerful empirical evidence that at the least the basic physical principles governing material objects have been effectively constant and unchanged over billions of years. We know that stars as far away as we can see produce exactly the same spectral lines as the Sun, so the chemistry is identical. In fact, if the laws of nature themselves changed substantially then we would not be able to accurately account for the processes in the cosmological past at all. So you have to admit that the laws are effectively unchanged as far as we can tell over cosmological scale. On what grounds, then, can one dismiss the claim that the laws do not change at all? That is, why believe the "strong form" of the principle that you advocate? One can speculate about the consequences of denying it, but you seem to think that there is some necessity to it. What is the basis of your confidence?

Cheers,

Tim

Dear Sir,

The Heraclitean position that everything real is in flux and the Parmenidean position that nothing real ever changes are not contradictory, but complementary. While the former refers to the state, the later refers to the result of observation / measurement. Because of our limitations, we and our measuring instruments measure different aspects of objects and their temporal evolution only in phases. Thus, measurement refers to the state at a given instant, which information is frozen for use to describe the object at other times. Thus, the Heraclitean position refers to the state and the Parmenidean position refers to the result of measurement. The intermediate position is also valid for the above reason.

Nothing in the Universe, except the Universe itself is unique. Thus, from the universal perspective (substantive part), the laws are unchanging. But from the perspective of the everyday world (application part), everything is ever changing.

Regards,

basudeba

Dear Lee Smolin,

It is a pleasure to observe your evolution toward 'realism', with one natural universe making up a unitary whole, existing in the objective Now, distinct from past and future. And a corresponding rejection of mysticism in physics, whether strings, or Platonic ideals outside time and space, or a multiverse. 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. As Tejinder Singh notes, it is an 'act of faith' at best! And I suspect it is far more harmful to physics than helpful. Pianos and banjos typically exist before Fourier, almost never otherwise.

You state that logic is the distillation of the fact that we can reason about (concepts). I see logic as the essential property of physical reality that allows structural AND-gates and NOT-gates, however implemented or constituted, to combine to produce ALL-gates, which are combined combinatorially in space and sequenced serially in time to produce, in effect, all mathematics and all control mechanisms, interfaced, where appropriate, to analog mechanisms. It is difficult to find an area of physics, from molecular to bio-molecular, to organic to silicon to neural, etc., where this sort of logic does not apply. Thus mathematics is eminently reasonable, arising from logical structure subject to time-based evolution, including the trigger events for counters composed of logical, real, physical subsystems that yield the 'next' natural number. This addresses Kronecker's "God made the integers, all the rest is the work of man." Counters exist naturally, from telomeres on chromosomes to crows counting; the Number operator is the counter at the heart of quantum field theory. I present a brief overview of such in my essay.

My essay addresses another intrusion of mysticism into physics -- the non-locality that Bell "invented", based on (what I claim to be) oversimplified physical assumptions. His math is correct, but physics that follows from false assumptions is not. I invite you to read my essay and very much hope that you will find time to comment upon it.

My very best regards,

Edwin Eugene Klingman

Dear Lee Smolin,

good essay for pointing out that human reasoning could have its own limitations. This would then also apply to our sciences, and last but not least to physics.

I pondered about wether your lines of reasoning to argue for naturalism could itself be somewhat a formal axiomatic system, hence be evoked by you (for whatever reasons).

My point here is to emphasize that we surely can try to give ultimate answers to our fundamental questions, but in any case - at least for me - it seems to be nearly impossible to prove those answers to be the one and only right answers. In other words, we cannot know with certainty that our assumptions and the resulting answers are facts about nature. We cannot even be sure that the human mind is priviledged (what is often assumed to be the case) to be able to come to those answers in a way that would leave no doubts. Therefore one had to prove the necessity of the used axioms and i think this would be just a similar task like proving the existence of God.

Dear basudeba,

In the sense of logic, the two positions are indeed contradictory, under the mild assumption that anything is real. Call some real thing R. According to the Heraclitean, R changes always in all respects. According to the Parmenidean, R does not ever change in any respect. These are contradictory.

Regards,

Tim Maudlin

Dear Sir,

Your ten year old photo is still you (R), though you are ever changing with time. Is this statement contradictory?

We have discussed it elaborately in our essay, which you are welcome to read. We will also read your essay and comment.

Regards,

basudeba

Dear Sir,

Mathematical reality is the quantitative aspect of Nature, which is logically consistent - hence unchanging - and harmonizes with other aspects. But the problem arises when we try to manipulate them. In one of the essays here, the final equation is consistent with the figures given. But if the same sets of figures are applied to the initial equations, it shows 1200 = -1250. The author has not cared to reply to our comment. With such basic flaws, even if the final equation turns out to be right, the theories become questionable.

Mathematical space-time structure is the intervals between objects that change according to the time evolution of those objects including motion due either to inertia (determined by energy at the point of application) or application of force (influenced by mass at the point of application), which change continuously due to interaction with its environment and determines the structure of free fall. Thus, the curvature does not belong to space-time, but to the position of objects that determine the interval. Geometry cannot tell matter how to move and in turn - only energy moves matter. Geometry is determined by such motion. We use alternative symbolism of such evolution of objects to describe the interval.

The Principle of no unreciprocated actions emanates from Leibniz's principle that there should be nothing in the universe that acts on other things without itself being acted upon. This is essentially Newton's third law. Einstein used this principle in GR when rejecting Newton's ideas about absolute space. His interpretation forbids any reference to a fixed-background and entities whose properties are fixed for all time, regardless of the motion of matter, thereby reducing interactions to relationships with other objects. But space itself is not only intervals, but also background for everything. All motion takes place in space in time. All observations are made at "here-now", which is referred to as space-time. Thus, by implication, Einstein's interpretation of the Leibniz's principle makes everything dependent on observation only. An object does not exist unless observed by a conscious observer - which concept of Bohr he opposed! Einstein based his conclusion on the M & M experiment, which used light. But light is a transverse wave, which is background invariant. Thus, his conclusion is faulty.

Chess was invented in India by the warrior class from their war moves. Like a poet writing poetry, the basic concepts were always there. But the processes of its codification for alternative use by emphasizing different aspects differently or choosing specific components, which can be infinite, were created. Your conclusions are identical with our essay, though the presentations are different. Best wishes.

Regards,

basudeba

Dear Lee Smolin,

The problem of inventing (invoking) mathematical objects in the development of a theory is that the invented objects may not exist in our physical universe. For example: the extra dimensions of space invoked in super string theories may not exist. To avoid this problem I used only the observed three dimensions of space and one dimension of absolute time to develop my theory Model mechanics.

Model Mechanics unifies all the forces of nature (including gravity). In addition it give rise to a new theory of relativity called IRT. IRT includes SRT as a subset. However the equations of IRT are valid in all environments including gravity. I invite you to read my essay and give me your informed comments. Thanks.

Regards,

Ken Seto

It is not contradictory to say I am changing and my photograph is not. I am not my photograph, and never have been my photograph.. It is contradictory to say of one thing that it is changing in all respects and not changing in any.

Dear Lee,

Nice essay from a popular author, having read your 'Three Roads to Quantum Gravity' right here in Africa years ago. You talked a lot about what exists in your essay and take a naturalist position on this, viz.

"...everything that exists is part of the natural world, which makes up a unitary whole"

"...all that exists is physical reality"

"... all that exists is part of nature"

"I would like to propose that there is a class of facts about the world, which concerns structures and objects which come to exist at specific moments, which, nevertheless, have rigid properties once they exist"

"Properties off mathematical objects, once evoked, are true independent of time"

"Both the records and the mathematical objects are human constructions which are brought into existence by exercises of human will, neither has any transcendental existence. Both are static, not in the sense of existing outside of time, but in the weak sense that once they come to exist, they don't change"

On the basis of these statements, I have a question for you and other members of the Perimeter Institute to ponder:

Can what exists perish? If (a part of) what exists can perish, what will be the implication for physics? Would this count as a fundamental event?

I discuss a postulate in my essay: the non-zero dimensional point does not have an eternal existence, but can appear and disappear spontaneously, or when induced to do so.

If contrary to belief, the point, either mathematical or physical falls among things 'which come to exist at specific moments' as you say and contrary to what you said 'once they come to exist, they DO change (and perish)', what then? I believe you have the clout to push this idea forward if you find that it has merit.

Best regards,

Akinbo

Belief is a religious concept. Physics and science are about usefulness that requires predictability. Metaphysics may be useful if it results in a hypothesis. For example the ``in the beginning..." of Hebrew tradition (LaMaiter, Friedman) or of the eternal universe of Hindu tradition (Narlikar, Burbidge, Hoyle).

Beauty and grace do not help us survive except as they identify useful relations.

I suggest ``discovered'' and ``invented'' are mutually exclusive. The ``evoked'' class is empty because the possibility of a game existed before the game. The sense of ``discovery, beauty and wonder'' has evolved to be useful. Discovery of a game means the discovery of a possible relation that is allowed in nature. Whether the game continues depends on its contribution to entropy (the selection process).

We are in search of a ToE. This search is for core principles. The trend is to find ever more basic principle (perhaps only for our limited ability to understand). Postulating that such principles exist has helped if only to get funding. The trend is toward fewer principles not more FAS's.

My paper suggests there are only number, geometry, and their relationship. Logic has evolved to further the relationship and increase the entropy rate. Math is a relationship that exists before humans.

Dear Sir,

Anything subject to time evolution must perish. Time evolution takes place in six stages: being (situation leading to its creation), becoming (its creation itself), (growth due to addition of other particles/events), transformation (as a result), transmutation (due to the same effect - incompatible/excess addition), destruction (change of form as a consequence) to start a new chain. This applies to all dimensional things. Since point has existence, but no dimension, it cannot perish. You cannot treat point as a small part of a line, as even the minimum unit of a line will have some length.

Regards,

basudeba

Dear Lee,

While I think LQG has a lot of truth in it, however, Dr. Tegmark is 100% correct. I proved that in my last essay and I will have much more evidence in my upcoming essay.

"Reality is nothing but a mathematical structure, literally"

FQXI article

more info

Dear Basudeba,

Thanks for your opinion. Perhaps, Lee would take both views together for clarification. I am of the opposing view that points have dimension of about the Planck size and can be treated as the extremities of lines, i.e. as a part of lines (see Euclid's definition in my references) but I will not force the issue as yet. You may also check the arguments in my 2013 essay.

You say point has existence but cannot perish. If the universe can perish and cease to exist, will all the points in it remain behind and not perish with it? Of course, I am assuming the correctness of the cosmology that the universe did not exist, started existing and increasing in size and will eventually perish.

Regards,

Akinbo

Lee,

пЃЉ Let me first say, my critical point of view was derived from the inspiration of reading your essay. It has ignited within me the motivation to address, in a separate paper, the many misconceptions we have regarding time. Hopefully we will debate my theoretical and hypothetical intuitions constructively and thereby provide a genuinely effective iteration of these conclusions at some time in the future (excuse the pun).

With your sections addressing the effectiveness of mathematics in mathematics and physics, you lead the reader into the assumption we currently understand the fundamental laws governing both studies and their comprehensive connections between separate deviations. Mathematicians are on a path to derive a cohesive fundamental structure connecting Algebra to Geometry to Logic just as Physicists are attempting to unify the laws constituting observable physical forces. Ultimately the goal will not only connect the separate entities individually, but also unify these distinct disciplines into a decisive understanding of nature. As my essay attempts to illustrate, mathematics is abstract, arbitrary, and purely a fundamental tool - we then apply it to a properly structured interpretation of reality - we describe this process as the discovery of the laws of physics. The idea that nature or reality can be described or mirrored mathematically or in physics is not a "mystical" conjecture of philosophy. Artists are capable of painting images observed in reality. In time, the art evolved discovering better techniques to make paintings an accurate description of observation. This evolution was brought about by the application of "discovered" techniques using fundamental tools (i.e. physics using mathematics as tools for observation). The limitation or effectiveness of their mirrored representation of reality depends solely on the applications of tools "invented "or applied by humans and our competent abilities discovered by the art form. Photography led to realism, Motion Pictures led to temporal realism and now we have 3D HD Projections which is more "effective" at capturing and describing nature, but we are not satisfied, we require more information to describe the enormous information needed for temporal realism. The information needed to recreate nature or a temporal event using Mathematics and Physics is infinite. We possess the necessary mathematical tools, Integrals and Calculus for temporal states, we simply do not have enough information to predict probability from possibilities, or we are too lazy to consider the enormous amount of information needed to accurately calculate and depict nature. Therefore our observations as mathematical representations are lacking in its description prior to an event, leading us to assume probabilistic and uncertain future events based on biased accrued approximations within our mathematical equations which lead to misinterpreted phenomenon or unexplained events. For the most part, we are simply satisfied describing nature to an acceptable limit, but without including every physical representation of the information concerning the system of observation, we cannot call this an assumption a direct interpretation of nature. It is lacking, but it is not the fault of the applied mathematics that already "existed" after which humans "discovered" and "invented" arbitrary imaginative symbols to represent. It's lacking, but not because of the interpretation of mathematics described as physics (although we must clarify a preferred theoretical methodology). It is lacking because we do not include enough information to describe an object "O", and therefore we have an abstraction we (as the interpreter) hypocritically claim is based on the limitations mathematics and physics pose on nature.

As unbiased investigators of nature (mathematicians, artists, theorists, and physicists), we must not assume a provocative or unconventional approach abstract in resolving a preferred method of depicting nature, is assumedly flawed or insignificant for discussion and interpretation until thoroughly proven otherwise by analytical means of exploration and experimentation. The approach necessary for discovery is a derivation of fact from fiction - truth from falsehoods - limited "gauged" probabilities from infinite uncertain possibilities we seek as discoverers of our reality.

All in all, it was a great addition to this discussion. Thank you for contributing.

Best Regards -Keep in touch!

D.C. Adams

Dr. Smolin,

Recent advances in neurosciences support the hypothesis of natural evolution of mathematics. Individual rods and cones in the retina, each only respond to simple a element of shape, motion or illumination. We seem to be hardwired to recognize what we call geometry, and evoke an ideal form which we mathematically abstract. That in no way compels us to presume that there exists an a priori Platonic realm. The reality may just as easily be that the universe is not perfect and that perfection of geometric form is an abstract of human desire that there be an absolute that would transcend our mortality.

"Most mathematical laws used in physics do not uniquely model the phenomena they describe." I can think of no greater instance, nor one more problematic, than the lack of a general concept for an existential definition of electric charge. Without such, we can not deduce the physical structure of what is perhaps the greatest natural wonder of all, the humble electron.

A toast to cottage country, jrc

Dear Sir,

A point is only a position in space. Space will remain in one form or the other. Otherwise there cannot be big bang or big bounce. Thus, point cannot perish.

Regards,

basudeba

I think I'm a temporal naturalist now!

"In closing, I would like to mention two properties enjoyed by the physical universe which are not isomorphic to any property of a mathematical object. 1. In the real universe it is always some present moment, which is one of a succession of moments. Properties of mathematical objects, once evoked, are true independent of time. 2..."

That's cool.

So, the first part seems to be a form of Presentism. Evocation is cool. I'm not sure in the next sentence that I would phrase it "independent of time" because the evocation is within the universe and the universe is within the present moment.

If there were more space it would be interesting to go into shape dynamics / time capsules / etc.