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

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 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.

Lee,

Your ideas on loop space with Astekhar and Rovelli in the 90s were fascinating. Your present ideas fascinate once again.

I think the Platonic view can be appropriately tweaked to agree with your view of a "unitary whole." Actually, I think that by default the view has always been that of the "unitary whole." I call it the view of an "all-encompassing existence."

My view is that there is one and only one totality of the existence. However, the one totality of the all-encompassing existence has two "initially" separate realms that evidently progressively get connected "inseparably" in the unified whole.

The following "table" illustrates my tweak of the Platonic view that involves the two fundamental realms (which are named in the header) into which the fundamental essences (which are listed under the header) are categorized.

The Realm of Phenomena ------ The Realm of Noumena

------------------------------------------------------

Space (the dimension) ------ Time (the dimension)

The Aethereal Substance ------ The Ephemeral Instance

Motion ------ Duration

The Corporeal Forms ------ The Abstract Ideals

------------------------------------------------------

My view is that the all-encompassing existence has both the phenomena and the noumena. The strictly phenomena being the corporeal cosmos, which we observe with its full spectrum that range between the kinematic vacuums of dark voids and the kinematic singularities of super dense black holes. We see within that range the electromagnetic spectra of phenomena and the particulate spectra of phenomena. As far as my view goes, the corporeal cosmos exists within the space dimension, and the corporeal cosmos is the space-occupying substance that is inherently rendered the kinematic definitions by the essence of motion. The space-occupying substance is aethereal if without the kinematic definitions; but it is never without the kinematic definitions (in string parlance, it always has branes and underlying branes ad infinitum).

Note that my Forms and Abstracts no longer follow the meanings of the Platonic terms. My "Corporeal Forms" are no longer the Platonic universals of "Abstract Ideas." They are now exact opposites. My Corporeal Forms are strictly categorized as phenomenal realities, while the Abstract Ideals are strictly categorized as noumenal realities. To me, both the phenomena and the noumena are real.

However, the corporeal is the more real than the abstract because it is the manifestation and embodiment of the ideals. In other words, the corporeal forms are the complete or thorough realities, the already connected phenomena and noumena, the embodied truths, the combined unified fulness of existential realities.

The laws of nature, the laws of motion that we try to discover, are the abstract ideals. In the abstract is our mathematics. In the corporeal is the execution of the mathematics.

My view is that the abstract ideals are static and only await their discovery or fulfillment. So, there is no evolution of the laws of nature. On the other hand, the corporeal forms are dynamic since the fundamental essence that defines the corporeal forms is motion (flux). Motion is that which is being constantly governed to conform to the ideals - to the laws of motion described by our mathematics. (Here of course is the bit of my tweaked Parmenidean and Heraclitean.)

The tweak that brings agreement with your idea of the unitary whole should now be obvious. As much as we understand, the mind, the nous, that perceives the noumena, resides in the brain-and-body that perceives the phenomena. We have the mind and the brain-and-body as the unified mind-and-body.

If the idea is extended in its application to a pan-cosmic or pan-existential view, the whole cosmos would be a sort of "super mind-and-body" - a unified whole of the mindset and the body-set that pursues the execution of the abstract ideals towards the continuous fulfillment of existential realizations in the corporeal forms.

Yet, the noumenal is apparently inherent in the phenomenal. The inherently unified corporeal-and-abstract reality is fundamental. There are the simple corporeal-and-abstract realities. And there are the complex corporeal-and-abstract realities. But, evidently, the simple corporeal-and-abstract realities are progressively and continuously transformed to form and sustain the complex corporeal-and-abstract realities.

It is apparent that the noumenal and the phenomenal may be established as a unified inseparably connected, or sustained, reality. All that is needed is an inherent and fundamental bias in the existence, in order to have an established cycle that sustains the connected reality. With a cycle limits are set, in which the excesses are spun off the sustained complex realities, and with the spun off fragments eventually grown into new complex realities. The spin off process actually looks like the emergent mechanism in the corporeal that bring about replication. Now, it appears that gravity is that necessary fundamental bias that is indicated in the mathematics of physics.

Lee, my submitted essay is more illustrative of the relationship between mathematics and physics, instead of being explanatory. But I have a book/ebook that is sold at my www.kinematicrelativity.com website and a few pages there explaining my work.

I have been focusing on the ramifications of the genesis formula that I discovered. I derived the genesis formula from the 3-d tensor transformation equation. I clarify in my work that the 1-d Galilean transformation implies mass increases, that the 2-d Lorentz transformation implies mass increases, and that mass increases because of the universal gravitational acceleration is implied by the 3-d Castel transformation (tongue-in-cheek, Lee).

The genesis formula implies that mass increases inherently and continuously occur for every mass domain in the cosmos, and hence for the whole cosmos, because of gravity that is a fundamental bias in the corporeal realm. The genesis formula implies a few other radical ideas.

A bit of an exchange of ideas and critiques between us would be nice.

Regards,

Castel

...cosmic mass-energy increases because of the universal gravitational acceleration are implied by the 3-d Castel transformation (tongue-in-cheek, Lee).

(gotta correct that..)