Essay Abstract

The effectiveness of mathematics in physics is reasonable because it is relative. We should reject the view, predominant in the history of modern physics, that mathematics offers a shortcut to eternal truth, and serves as the oracle of nature and the prophet of science. We can better understand mathematics as an exploration of a simulacrum of the world from which time and particularity have been sucked out. The radical selectivity of mathematical reasoning helps explain its power, its limits, and its dangers in physics. Mathematics is good at some things, and bad at others, especially at those that are historical. That limitation matters if the most important fact about the universe is that it is what it is as a result of its history. Mathematics cannot replace physical intuition or empirical discovery. In return for the immense service that it renders, it offers science a poisoned chalice: the idea of immutable laws of nature.

Author Bio

Philosopher, social and legal theorist, and political activist. Author, most recently, of The Singular Universe and the Reality of Time, with Lee Smolin, in which each of us presents separately the entire argument of the work. Professor of Law, Harvard University. Member of the American Academy of Arts and Sciences. Minister of Strategic Affairs in the government of President Lula, Brazil.

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Dear Mr.聽Mangabeira聽Unger,

Thank you for your enjoyable and well written essay.

You mentioned that "Mathematics gives us no royal road to truth about nature", myself share also this view. Math may intersect with physics and in some cases only in rough manner. The truth is that part of the math world can be unphysical similarly the physical world can sometimes be non-mathematical.聽

We have tangible circumstances in the physical world, i.e. where life is generated in particles and this聽phenomenon聽doesn't fit into math at all, I have touched many more examples of this sort in my article. One essential fact the bridges us to math and physics is "quantity" that can be measured, anything else in our world not obeying this rule is omitted as unphysical, this is one of the big dilemmas of contemporary physics.

Warm regards

Koorosh聽

Dear Mr. Mangabeira Unger,

It was a real pleasure to read your essay. It was interesting and accessible.

It opens up questions. For example: when you highlight whether any mathematical construction will have no assured application to the real world.

I have a different view on mathematics and time. I might be wrong but time exists in mathematics. Probabilities are inside time. Could probabilities exist without time?

In my view, the representation of time is different in mathematics. In probability, time is represented as a discontinuity. I find this property interesting.

Regards,

Christophe

Dear Roberto Unger,

In your essay you state that "causal explanations make no sense outside time; causal connections can exist only in time." This I agree with. But then you say "...the moves in a mathematical or logical chain of argument do occur outside time." I'm not so sure of that. A mathematical argument goes from step to step in sequential fashion which seems to incorporate the nature of time. As it does not matter when one steps through the sequence, it is time-independent. Much of physics (the physics covered by energy conservation) is time independent: dH / dt = 0. I do not see this as equivalent to the Platonic vision which does truly seem to proclaim a realm outside time and space.

In similar fashion I view logic as a property of reality that allows the physical structure of AND-gates and NOT-gates. The physical implementation of logic gates, combined combinatorially in space and typically sequenced in time, provide counters that generate the natural numbers and address Kronecker's maxim: "God made the integers, all the rest (of math) is the work of man."

Thus I see logic not as an "outside" rule or "law" but as a (the?) primary property of physical existence, supporting a single, self-consistent, unitary reality. Physical evolution in time yields math 'circuitry' at almost all levels, but perfected at the level of man. The logical operation of such circuits (in a computer, a cell, or in our brains) is independent of time in the sense that it does not typically matter when the logic sequence is triggered nor how long the steps take, but still, the physical existence and operation is embedded in time. Of course structural changes that 'endure' in time record information, and this too is typically time-independent, but is in no way 'outside of time'. Thus all the basis of math is derived from and 'evoked by' physical reality. This operation of the universe is not "subject to laws" outside time, but we can abstract relations (as I briefly show in my essay) that capture the operations reliably and thus appear to have the character of law, or "timeless truth" -- probably more accurately stated "time-independent truth."

Finally, I fully agree that "mathematics cannot replace physical insight." As an example I show in my essay how mathematics, based on faulty physical insight, led Bell to introduce a mystical 'non-locality' that almost banishes physical insight. And this is not the only 20th century mathematics that muddles physical thinking. I see the correction and clarification of these induced mystical concepts as the greatest need in today's physics. Then we might move forward. Most movement today impresses me as lateral or even backward.

You certainly could have been thinking of John Bell when you stated:

"The less we grasp the non-mathematical reasons for the application of mathematics ... the more enigmatic and disconcerting the application of mathematics will appear to be."

Bell did not grasp the underlying physics, and thus based his mathematical treatment on false assumptions. The correct application of mathematics to incorrect physics has certainly led us to enigmatic and disconcerting conclusions. In this sense mathematics is as you say, "a good servant but a bad master."

Thank you for your essay and I invite you to read and comment on my essay.

My best regards,

Edwin Eugene Klingman

Dear Roberto Unger,

To be sincere I am wondering what exactly is going on here? I look forward to the book you are co-authoring with Lee Smolin to know exactly how the course of scientific truth will be advanced. If you have time, you may take a look at my essay and comment.

Regards,

Akinbo

Dear Sir,

Mathematics is the quantitative description of Nature. It is linear (explication) or nonlinear (including recursive reasoning) accumulation and reduction in numbers. Number are concepts that differentiate between similars - if there are no similars, it is one; otherwise many; depending upon the sequence of perception of one's. Being a concept, number has no physical existence (other than the objects to which the numbers are assigned for the moment) - thus no time evolution - it coexists with time. "Mathematical and logical reasoning takes place in time" because the concept of numbers are associated with physical objects that are subject to constant time evolution. Then why do you say mathematics suppresses time? Or did we misunderstand your statement?

The controversy regarding whether mathematics is discovery or invention is unproductive debate that can be overlooked. Every living being - animals included - have at least some primitive mathematical sense (up to number three). Higher number mathematics is like unexplored territory - it can be discovered one area or much more at a time. It is the invention view that has created the problem through trial and error methods. The danger is in "bootstrapping of an activity that is turned outward to nature, viewed under a particular aspect". Often this leads to reductionism of the worst kind. There is a story about six blind men, who went to see an elephant. Each touched only one part of the creature - leg, trunk, ear, belly, tooth, tail - and described the elephant by that experience only. Though all their descriptions are valid, one who has not seen an elephant cannot make any sense out of their combined statements.

Then there is the problem of as you put it - "multiplication of equivalent propositions be mistaken for the marking of synonymy". As you say: "the effectiveness of mathematics in physics is reasonable because it is relative". However, when numbers are divested from physical representations, it leads to other problems. 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. In our essay, we have given many examples of why theoretical physics has gone astray with many contradictory interpretations.

The problems of explaining "change and puts structural analysis in the place of causal explanation" arises because of our improper understanding of change. Changes are of two types: an object with an internal structure evolves in limited ways based on changes of its mass energy content (all classical objects and quantum objects up to nucleons). This type of change is not applicable to fundamental particles. They only change their position in time.

Regards,

basudeba

Dear Prof Unger:

You will not recall, but I once attended a seminar you gave. That person who kept asking what you mean with "real". I will not bother you with that again for I know it's an unfair question and in a way I was pleased you didn't pretend to have an answer. Let me thus get straight to the point of my essay and how it touches upon the topic of yours. You write:

"Mathematics deals with nature as well as with itself."

I've never seen "mathematics" dealing with anything. We deal. Note the difference. It doesn't matter whether you believe mathematics is invented or discovered. I find this a pointless discussion. The relevant point is that WE use mathematics FOR science. But what if we find out that mathematics has limits? Can we still do science? In my essay I argue that yes, we can, and that we already do.

-- Sophia

    Dear Madam,

    Reality is whatever exists (has a confined structure that evolves in time and is perceptible), is intelligible (perceptible/knowable) and communicable (describable using a language as defined in our essay). Number is a perceivable property of all substances by which we differentiate between similars. If there are no similars, it is one. If there are similars, it is many. Many can be 2,3,..n depending upon the sequence of perception of one's. Mathematics is the quantitative description of Nature. Thus, it explains only one part. Another part is described by physics, which has meaning only when observed (perceived).

    Thus, mathematics can be figuratively said to deal with quantitative aspect of Nature and because of logical consistency, deal with itself. However, since it is we, who perceive the numbers - hence mathematics, we agree with you that we deal with it.

    Regards,

    basudeba

    Lots of physical-mathematical activities and perceptions doing the rounds.

    Great, way to go!

    Respectfully,

    Miss. Sujatha Jagannathan

    22 days later

    Dear Mr. Unger,

    Congratulations. Mathematics is indeed a misrepresentation of the real world if it claims to describe the whole as the sum of its parts. There is no evidence (upon which mathematics relies) that nature shares in the timelessness of mathematical propositions. Change is the only constant in nature - a provocative idea when applied to the pursuit of mathematical truths.

    In question is whether or not there is discontinuity between natural fact and mathematical truth. We represent the former with words and the latter with numbers. In the world of nature truth is transient while in mathematics facts are represented as being true and constant. Numbers don't lie unless they are improperly related, as is the case when they are intended to "trick" others.

    I identify with your "most decisive and astonishing feature of mathematics, and the one by virtue of which it can be neither invention nor discovery, as they are conventionally understood. This trait -- the fourth attribute of mathematics -- is the study of a counterfeit version of the world, of the only world that there is". Mathematics is a distorted representation of the world because it only addresses those aspects of nature that can be enumerated. "Deception" is a heavy word but, used intentionally or otherwise, mathematics is correctly identified as such.

    Mathematical reasoning is a self-chosen format for inquiring into the world whereby we brush away any unmanageable considerations, e.g. time and phenomenal variation in nature, in the interest of gaining certainty. However the only certainty that is so-gained is that mathematics does not represent nature the way it actually is.

    Gary Hansen

    Dear Professor Unger,

    In your essay you state many important truths. Things in nature are thoroughly time-bound. We ourselves and our thinking processes are not exceptions. Mathematics cannot describe or represent all aspects of time, and therefore it is a serious error to suppose that the world is fully mathematical. With all this I agree.

    My concern, however, is with denying that mathematical truths, some of them at least, are in their own right timeless and eternal. Insofar as such truths are exemplified in nature, then some truths relevant to nature are also timeless. You seem to acknowledge this with your example about syllogistic reasoning. As you say, the logical relations are timeless, although human perceptions and uses of those relations are acts which occur in time. Granted, if we allow for some timeless reality, then we have to explain the connections between the timeless and the temporal. As you say, Plato did not successfully provide a bridge between the two, and neither have later thinkers solved the problem.

    Nonetheless, if we deny timeless reality altogether, maybe we face problems that are worse. Consider, for instance, truths about prime numbers. You maintain that these facts are about the natural world, the only world that there is, although they are second-order or higher-level facts. If that is so, then what happens when the world changes so radically that the underlying facts, even the laws of nature, are different? On the view that time is the deepest reality, I think the indicative "when things are different" is preferable to the counterfactual subjunctive "if things were different". Sooner or later, things will be different, and things were different in the past. For the prime numbers there would appear to be only two alternatives. Either the facts about the prime numbers will sometimes also change, or those facts will always stay the same. Suppose that the facts about prime numbers sometimes change. This supposition is contrary to the standard way of thinking about prime numbers. A more serious objection is that, if there is a dependence of mathematical facts on underlying physical facts, then we should be able to develop some conception of the changes in mathematics which would result from various types of changes to physical reality. But we do not seem able to develop any such conception. Suppose, on the other hand, that facts about prime numbers remain the same, no matter how much physical reality changes. This is, I think, the more plausible alternative. In this case, the facts about prime numbers are independent of all temporal changes. In that sense these mathematical facts are timeless and eternal, and many people would also call them necessary. Some of these facts are "startling", but they possess something more than the "it-just-happens-to-be-that-way character of ordinary natural facts".

    Although I am not able to agree with you about mathematics, I do agree with your insistence on the importance of time. I appreciate your stimulating work on these topics and on other topics.

    Laurence Hitterdale

    10 days later

    Dear Professor Ungar,

    I posted a comment at your site that was unnecessarily contemptuous and devoid of the civility all contributors are entitled to. I deeply regret having done so, and I do hope that you can forgive my slurring of your fully deserved reputation.

    I suspect that I may be suffering a relapse of Asperger's Disorder. While this might explain my distasteful action, it cannot in any way justify it.

    Respectfully,

    Joe Fisher

    12 days later

    Dear Professor Unger,

    I thought that your engrossing essay was exceptionally well written and I do hope that it fares well in the competition.

    I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

    All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

    Joe Fisher

    19 days later

    1. Dear Roberto,

    I read our essay with the ease that a clearly flowing exposition adds to finding ideas congruent with my own.

    The aim of my essay was to outline a formal system that is consistent with much of what you argue for, but looking back at it, I see it is probably very hard to follow the thread of too many ideas packed into a few pages.

    Here are some of these strong congruences between your terms and my essay (搂聽6):

    ``The one real world'' -> the universe is the set of all perceptible events at a `given moment'. (All that is accessible, by any means.)

    ``Particularity'', ``individual phenomena'', ``time-bound particulars'' -> events are fixed entities of space-time.

    ``Time holds sway'' -> time is defined as memory in my framework, that is, time is observed (even measured) only through memory. Memory is sedimented events, sedimented time. Sedimented events are what we call space, or structure. This definition works perfectly well to describe any basic cognitive operation, which in elementary form consists in connecting two memories, and up to describing any structure --that we may term mathematical or not.

    In this formal framework, the `evisceration of time' you speak about is clearly and naturally displayed.

    It does so by including cognition, which could be a natural way to expand your framework:

    If you define symmetry as immunity to a possible change, you have the seed of a very apt concept to describe perception, as well as any law in the most general sense: A perception is a wager, a bet; perception works because when you encounter a situation, beyond being completely unique --particular--, it has changed aspects, but also aspects which are conserved, from previously known situations, and so you decide how to judge, to behave, to act. Hence a perception is a symmetry. It is inherently a categorisation. It is also necessarily a simplification, a reduction of the world. It is also an analogy, and all these terms can be seen as occurrences of symmetries. By building classes, or categories, perception `averages' on the whole class, hence a perception is a smoothing of the world. This is the only way to be able to predict, in a set of a priori particular, singular events (because prediction supposes some sort of recurrence, of reproducibility, both again are symmetries). So with symmetry we have a general expression for any perception, and any sort of law, included those we derive in physics. Included, also, any mathematical structure.

    This formalisation clearly shows how me move from particulars to laws, or structure, or predictions. So moving out of particulars is in the very nature of a prediction, a concept that is evidently so pivotal for a living individual --were it strongly cognitive like humans are, or not: to live is to display a know-how, an ability to de facto predict correctly.

    2. As a consequence, we also see that a perception operates under a particular interpretation, that is, a structure that you endow this singular situation with (singular situation that I represent by an a priori completely amorphous set of events).

    This is exactly how I understand your beautiful expression of second-order facts, in a precise sense.

    Perceptions are second-order facts; they are interpretations, they are formed in the frame of reference of the subject, the living being whose survival is engaged in perception-action loops. And the basic criterion for `success' is that the bet that each such perception, each such interpretation is, does not lead to death, so that the game can go on.

    Interpretations are always made in the reference frame of a subject. Moreover, a cognitive subject is only able to make interpretations, never to express anything absolute about the world. All his statements are relative to himself, and his own experience. All we can express are interpretations about the world, not first-class knowledge.

    A statement like ``the world is mathematical'' is absolute, it affirms something ontological, first-class, and as such, it does not comply to Popper's criterion of scientificity. Whereas taking it as a working hypothesis, to see the consequences, is acceptable. It is striking that so many physicists fall into this trap.

    The same can be said about most presentations of the idea of a multiverse, and, of course, of extreme Platonist positions according to which the mathematical realm is completely disconnected, pure and independent of our world. Or are non-scientific because they are absolute, they purport to an act of faith. And, made apparent when we include the cognitive subject in our formalisation, they deny that knowledge is relative to a cognitive subject; that, in a strong sense, a subject is a reference frame for knowledge.

    In passing: when you say that according to one thread of ideas, mathematics is discovery, perhaps `recollected', is it an allusion to Plato's conception of knowledge?

    It has in common with the idea of multiverse that it naturally tends to completely unfold all that is conceived of, even remotely conceived of, into a single, static space (devoid of time).

    The crux is that this unfolding is of little use. What do you gain once you say that there is a set of all the possible chess games, and evaluate the absolutely huge combinatorics implied? There are more illuminating ideas in Borges's Library of Babel, which, perhaps, contains all the possible books of a given size, included the one containing the true theory of the library, and many others differing by a single character, and all its possible refutations: all this profusion is of no use, past the profitable head-spinning phase that the perspective provokes. And that realisation is already present I believe in the old Hebrew tradition, when it says that everything that can be expressed is within the possible combinations of the 23 letters of the alphabet.

    To return to the multiverse fiction, if I use my definitions, `unobservable universes' is just an oxymoron. Good for literature or poetry, bad for a built, scientific knowledge.

    Reading Smolin's essay, I wrote down a question that I eventually did not ask in my comments. It ran: As much as you dislike a Platonic position, that is to say, a static vision of the (mathematical) world, I suppose you will not either like a multiverse theory, which in the same way unfolds a sort of full combinatorics of all the possible worlds? I suppose my interpretation was right, considering the closeness of your positions.

    3. I feel close to you again when you raise the idea that our concept of time is closely related to the issue of the effectiveness of mathematics. I can certainly listen to physicists who claim that time is nonexistent, if that is a hypothesis they enjoy (Einstein is famous for having insisted that time is an illusion). But firstly they certainly cannot state that in an absolute way --which they dare do. Secondly, how fruitful is that hypothesis? It is certainly sterile, in that it does not speak at all about all our experience, without any exception: events cannot be removed. Things done cannot be undone.

    It is worth quoting (again, since I have inserted it in the comment on another essay) an interview of Heisenberg in

    Glimpsing reality: ideas in physics and the link to biology Paul Buckley and F. David Peat, editors University of Toronto Press Revised edition (March 1996)

    ``PB How does quantum mechanics deal with time flow or does it in fact say anything at all about it?

    I would have to repeat what C. von Weizs盲cker said in his papers: that time is the precondition of quantum mechanics, because we want to go from one experiment to another, that is from one time to another. But this is too complicated to go into in detail. I would simply say that the concept of time is really a precondition of quantum theory.''

    What I say is that all what we live is under the seal of irreversibility, which means exactly that past events cannot even be accessed; however, all the matter of cognition is to use the past to bet about the future. This is possible with memory, by retaining a record of events, that can be reversibly accessed (that is, you can read again and again the record of any event, at will). Which means that you can try arbitrary connections between events. By turning time (always occurring events) into memory (or space, or structure, or writing: all forms of sedimented time), reversible access to images of events is possible. The whole matter of cognition, of reasoning, is reversibility. Hence the deep involvement with time.

    In this view, the ``basic asymmetry in the relation of mathematics to space and to time'' that you raise appears as a symmetry breaking inherent in sedimenting time into space, by leaving traces of events in the world.

    4. I have slight reservations, on what I feel are good points that you make, but without completely going to the end of the argument.

    You say that mathematical propositions are `outside of time' and `removed from nature' (p.聽4) quite convincingly, but you expose yourself to the criticism that this eventually means nothing different from a timeless, independent realm of extreme Platonism. So a Platonist would say that you just seem to mean something different. The same applies to your denying mathematics being `objects of a special type' (p.聽5), which mathematical objects definitively are, in your own words --though again I agree with what you mean.

    I have not been able to find a convenient answer to this attack, placing myself within the confines of your essay. While I agree with what you mean, I cannot use your text as a demonstration that I could reuse to convince someone else.

    On the particular point of the relation to time, I believe that the formalisation I have just restated (hopefully, in an understandable way) is very fruitful to position cognitive elaborations and the world in a common theoretical framework.

    5. Three small points.

    I have a question聽: when you write ``The enigma solved: the effectiveness of mathematics in physics is reasonable because it is relative'', what is it relative to exactly?

    While I understand the connection you do between recursive as you explain it and taking ``itself for a subject'', the latter is more appropriately called reflexive, and the concept is much larger.

    I would avoid invoking causality as you do, a concept which can only blur your text, since as I understand it it is completely unnecessary to your explanations. I have tried to explain why it is tricky to invoke causality in my comments to Lee Smolin's essay.

    Regards,

    Dear Roberto,

    It is certainly true that, as you are saying, "mathematics ... addresses a nature from which time and, together with time, all phenomenal distinction have been sucked out." However, your conclusion that "mathematics gives us no royal road to truth about nature" contradicts to the tremendous success of fundamental physics. Your essay misses the questions about the fine-tuned universe, why there are life and thought in the world and why the world is so impressively theoretizable. Simplicity and "naturalness" of the laws of nature cannot explain why they are so simple as to be cosmically discoverable and at the same time why they are open to a possibility for living and conscious beings to emerge. On the other hand, I cannot but agree with you in what I would call an abuse of mathematics, or, wider, an abuse of reason, to use Hayek's term. Physics is mathematical, and there is nothing but nectar in that (rather than poison). What is not nectar, is absolutization of that approach to all of reality, which already lead and continues to lead humanity not only to epistemological mistakes, but to massive tragedies.

    Best Regards,

    Alexey and Lev Burov.

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