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

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 Tim Maudlin,

I would like to comment on the interesting objections you raised. Let me start by the end, when you asked what reasons there could be to believe in such "strong form" of the principle. I think one reason is the following (which I ignore if it motivates Smolin, but it did motivate Peirce): in the mixed view, the stable principles that never change might be considered puzzling. One could ask the questions: What is the ontological status of those principles/laws? Why there are these laws and not others? And, irrespective of their form, why there are principles/laws at all?

Of course these questions might be answered without changing the traditional view of laws, but a legitimate possibility is to substantially change our conception of what we take laws to be. As you know this has puzzled very little physicists but has been widely discussed by philosophers, some of which have defended "deflationary" views on laws (e.g. humeans, antirealists, or those that postulate primitive causation instead of laws). Unsatisfactory as these accounts might or might not be, the point is that there are reasons to be skeptic of the traditional view of a constant set of governing laws. Then, the view that those laws also change becomes interesting to explore. At least, for those that find puzzling the traditional view on laws (and if I'm not wrong, you are not on this side).

Besides this observation, I would like to suggest a way out to your argument against the change of laws (i.e. the empirical evidence suggesting the lack of change of the actual laws of physics).

A way out would be to put forward a positive answer to the question:

Are there accounts that explain how from an underlying level guided by changing dynamical principles constant behaviour nevertheless arises?

If so, under what conditions? Of course, the conditions must not assume any sort of constant dynamical principles...

Accounts of this sort, if they exist, would help to harmonize the heraclitean view with the lack of change of the actual laws of physics.

In such a scenario, laws could be changing in a fundamental level (they should not change according to a meta-law, because therefore little would have been gained) while stable regularities would be exhibited in higher levels and codified by the actual non-changing laws of physics.

Best regards,

Aldo Filomeno

PS: I would also like to say that there is not really a problem of logical consistency due to the self-application of the principle (unlike the liar paradox). The reason is just because, under a charitable and intuitive reading, 'everything' just does not need to refer to itself (as well as it is not Smolin's intention to refer, say, to mathematical truths). Roughly stated, it refers to every thing in the world plus to the laws that describe the behaviour of every thing.

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".

Dear Lee Smolin

I could never accept that math is only what is defined with axioms, but I think that math is predefined without axiom. But, here I do not think so a Platonistic world, but a Physical world. Thus naturalism agrees with my intuition.

I agree a lot of with your ideas. This your ''evoked'' can be described a little differently. A lot of options in Platonistic world, for instance chess rules, mean a big entropy, which cannot be applied, similarly as perpetum mobile of the second kind cannot used heat if temperature differences do not exist.

Beside of this entropy, here is also time, which should to run, logic needs time, and consciousness which should think about this Platonistic space. Thus Platonistic world exist on some way, but it is ineffective, similarly as you described with your ''evoked''.

One essential distinction between you and me is, that I am reductionist, thus I claim that math of fundamental physics is simple. (It is also one essay about this in this contest.) Admittedly, special relativity (SR) is a correction of Newtonian physics (NP), but Newtonian physics also gives some information; SR does not mean complete correction of NP. Thus I also claim that quantum gravity should be simple.

Admittedly, transitions from Newtonian physics (NP) to special relativity (SR), to General relativity mean complication. But, c at SR gives connection between space and time, and G gives new explanation for gravitational force. Besides, SR is not so complicated as we think, a lot of is our familiarity with NP. One special example is the complexity of quantum field theory, but it is a consequence of quantum gravity, thus I suppose that it should be simple, as it is written in my essay. Postulates of SR and GR are simple, and it seem by Zeilinger, that postulates of quantum mechanics can also be simple.

You worked a lot of on this question, thus you can give a lot of anti-arguments. I am interested in them.

Thus, I claim that mathematics is a consequence of physics, but physics is also a consequence of mathematics. I gave two opposite examples in the second section, Planck spacetime and Pythagora theorem.

You even prove that Feynman and Mermin were wrong with ''Shut up and calculate'' and ''philosophy is not usable in physics''.

You mentioned that ideas are not massless. I mentioned in my reference [1] that time and mass are connected in dimensionless quantities, thus this makes your idea still more clear.

But, the problem is also to use right words. These essays gave a lot of improved essential sentences.

Dear Lee,

I enjoyed reading your essay as the concept of naturalism is an idea that I had not previously seen. It does seem to be similar to realism and I support this viewpoint.

One comment in your essay surprised me that in your definition of 2. the inclusive reality of time, you state that there are no eternal laws. Do you mean by this that there are no laws that can be expected to operate in the same way at different times?

In my essay 'Solving the mystery' I concentrated on the cases where there is a mystery in the connection between Physics and Mathematics and took the view that it is our lack of understanding that leads to the apparent mystery.

Richard Lewis

Hi Lee--

Your essay is outstanding: well-written and well-argued. Naturalism is the only way to go, and your interpretation of the role of mathematics is spot on (i.e., as a way to "summarize the content of records of past observations"). In particular, I appreciated your Conclusion, in which you highlight two properties of our physical universe that are not isomorphic to mathematics.

Personally, I've always been mystified by physicists who adopt Mathematical Platonism. I wonder about these guys: Do they not, like me, use analytical simplifications, numerical approximations, linearizations, and perturbations (to name just a few techniques), every day and in every way, to make progress in physics? Just how isomorphic can mathematics be to the physical world if physicists must typically rely on such mathematical techniques to get the job done? Put differently, if "A Supreme Something" had ordered me to design a physical world--and to do so in way isomorphic to mathematics--I'd like to think that I could have concocted a physical setup far more computationally efficacious than the one we now found ourselves in!

I have two questions regarding your excellent essay:

First, regarding the reality of time, you argue that there are no eternal laws. In taking that position, how influenced were you by John A. Wheeler's article, "Law without Law"? Back in the day, when I first read it, I am embarrassed to say that I thought Wheeler was crazy. In the fullness in time, I now see the wisdom in his position. Hence, I was primed to agree with your take on time today.

Second, regarding our singular universe, you write: "All that exists is part of a single, causally connected universe". In taking that position, do you necessarily imply that the universe must be both singular and finite in scope? Put differently, in an infinite universe, how is it possible to have a single, causally connected whole?

Best regards,

Bill.

Hi Lee,

You said, "In particular, there is no mathematical object which is isomorphic to the universe as a whole, and hence no perfect correspondence between nature and mathematics." As an example to defend this viewpoint you said, "In the real universe it is always some present moment, which is one of a succession of moments. Properties off[sic] mathematical objects, once evoked, are true independent of time."

If the universe was known to be discrete and finite at any given moment of time, (and time itself was also discrete) would this change your view that there is no mathematical object that is isomorphic to the universe? Do you think if we saw the universe as a computation that the notion of "time" might stand out more? Could time just be the step where the system/universe is in the calculation?

You said, "The alternative to believing in the timeless reality of any logically possible game or species is believing in the reality of novelty. Things come into existence and facts become true all the time." Is this analogous to adding an axiom to a math system?

With regard to this view, what are your thoughts on the following two statements?:

2^192748098245-1 is prime number.

2^192748098245-1 is not a prime number.

From your point of view, would you say one of these statements will be "discovered" to be true, but the truth of these statements did not exist before the notion of numbers, primeness, and the meaning of these symbols were "evoked"? Does "evoke" relate to something being defined or given meaning? Does this relate to how the meaning of a string of symbols created by a formal axiomatic system should be looked at as being devoid of meaning? (e.g. See Douglas Hofstadter's discussion on his p-q system)

Do you think metaphysics plays no role in science? Can we not know something about chess, haiku, or the blues even before they are invented if their rules which govern their structure/definition fall into a certain class?

Your statement that "honest wonder about our world seems a better stance than mysticism" seems to contradict your notion of "novel" events being evoked. It seems like if these things are really "novel"/random there wouldn't be a reason that explains why they came into existence, so at that point we could stop wondering about the reason and just be mystified by their existence.

Thanks in advance for your comments on these thoughts, and please check out my Digital Physics movie essay if you get the chance. There are some questions posed at the end of the essay that may interest you. (e.g. Do you think there is an analogy between the following relationships: a "class" vs. a "set" and "true" vs. "provable"? )

Thanks,

Jon

Lee,

As time grows short, I am revisiting essays I have read to see if I have scored them, and as your scoring reflects, it is one of the best. Hope you have the time to look at my essay in the remaining time.

http://fqxi.org/community/forum/topic/2345

Thanks,

Jim

Dear Lee,

I agree that mathematics comes into existence for us when the games that are its definitions become sufficiently clear. It is, however important to realize that mathematics is quite unlike a game with set rules, more like an evolving panoply of interrelated games undergoing an evolution. Once a structure becomes sufficently well-defined, it gains a relative Platonic existence in that it is independent of any one of us and can be investigated as though it existed independently of all of us. It is better to speak the way you do speak than to imagine that all of mathematics has always existed in a Platonic realm. The notion of games and definitions is very seductive however, for we are actually dealing with articulations of concepts and what we can manage to articulate often does have a history reachiing way back before the event of definition. Consider knots as a study and we admit that knots have been in our culture and our empirical understanding for thousands of years. Only recently did they attract mathematicians attention, and then at first only for their topological properties. In a real sense, these topological properties had also existed in the properties of real rope for those thousands of years, but mathematical models only emerged in the 19th and 20th centuries. Now slowly we begin to find out about the evident physicality of knots with more difficult models. We might someday find the minimal length to diameter ratio for the simplest knot! (a quantity implicit in the uses of rope) Concepts like the Platonic worlds are better seen as limits of actualities of existence of concepts in relation to our actions and evolution. I imagine I am saying things that you agree with, but am curious about your reaction!

Best,

Lou Kauffman

Dear Lee,

Here my comments, questions, and suggestions about your essay.

Contents

1 A pleasant essay.

2 Platonisms, and making one's ideas clear.

3 The explanatory power of a formal system.

4 Some room for improvement.

5 A path to improvement.

6 A Question.

7 Two quotations.

1. A pleasant essay.

It is a pleasure and even a relief to find a contribution by an accomplished physicist that integrates time as something substantial.

It pleases me of course because it is closer to my views than many others. I even have the feeling that it is congruent, it combines well with what I have tried to express (in particular, a central rôle of time, as we experiment it, `irreversible').

The essay is well written, it is easy for the reader to follow the author, and... it answers the question, which is worth mentioning since not all essays do.

Most of the statements touch on things that I feel right and I am sympathetic to. All the pieces fit well overall to form a pleasant discourse. The result is then that we ask more from it, we want to put that framework to the test, we want to try using it.

2. Platonisms, and making one's ideas clear.

The form of Platonism that you attack --aptly I believe-- is quite extreme.

Once I witnessed on one occasion Jean-Pierre Changeux trying to take the upper hand against Alain Connes in a conversation, by treating him of Platonist, for the mere reason that he had said that the solution to a mathematical problem is fixed, you can do whatever you want, it won't change, and that's a characteristic of mathematical objects. What Alain Connes was saying is just what everybody has to agree about, as you write: ``if any person can demonstrate it, any one can'', these are ``properties which can be discovered or proved, about which there is no choice''. Mathematical objects show like inescapable invariants.

In this situation, you would have been termed a Platonist. As usual, not much interesting can be said unless we define precisely what we mean by each term we use, and that implies getting closer to a formal system, and giving pragmatic criterions to define the key concepts, so that someone else can virtually reproduce, at least mentally, each experiment and ascertain semantic correspondence with what we signify.

3. The explanatory power of a formal system.

About formal systems, everyone is free to decide what rules he wants. Only, some decisions are really unfruitful.

One way of being unfruitful for a formal system is the inability to explain some classes of facts, by its inherent structure. In the present case, of course, if you make `mathematics' and `the existing universe' two completely separate domains, by construction, you have difficulties articulating them: there is by construction no natural connection between them. That's in summary what happens to Wigner. It is merely a consequence of the axioms.

Only should one be conscious that it is useless asking unanswerable questions, in a given system. So, more specifically, some decisions about a formal system are unfruitful in that they prohibit specific explanatory schemes.

Of course, even while a Platonist states the axiom of complete separation between `mathematics' and`the existing universe', he does enunciate the existence of both, and hence their connection, if only through himself because he names them, he has ways of connecting them, of interacting with each one, so he does recognise they both belong to a larger world. Most of the ensuing confusion stems only from occasional shifts to the general discourse in which you nested your formal system, when you should only be talking from within the formal system itself. If you stick to our axioms, you cannot ask about the effectiveness of mathematics in representing the world. If you want to explain the effectiveness of mathematics, you need a general, abstract domain in which objects that you observe in the world and objects that you meet in mathematics are both produced from a common root, and hence display natural symmetries.

Incidentally, the extreme Platonism you oppose to would fail to pass the test of defining pragmatically the terms it uses. That's another failure.

4. Some room for improvement.

First, you include causality as a constituent part of your principles. Causality is a human concept, in the best case, a theory. To me your sentence would sound better to me ``All that exist is part of a single, connected universe''. Or Simply ``the universe is everything''.

It is worth avoiding causality at this stage, because it is indeed absent of most of (the formal part of) physics. Let us take an example as simple as possible, to avoid being distracted by technicalities: Newton's law F12 = -F21 is completely devoid of causality. It says that the forces between two objects are permanently --and sort of magically, almost fortuitously-- equal in modulus and opposed. According to the equation, things, if they happen to change, change simultaneously; therefore, not causally. Writing an equation means suppressing any idea of causality. In spite of that, most explanations accompanying this and other equations do speak in term of causes, included in physics textbooks. You then have a complete strangeness between the actual theory and the discourse seemingly explaining it. See, e.g. John Clement. Student's preconceptions in introductory mechanics. American Journal of Physics, 56:1 (January), pp. 66-71, 1982. Or, even better (but in French) Laurence Viennot, Raisonner en physique : la part du sens commun. De Boeck Université, 1996.

You may like the following quote, which I I understand well matches you notion of being evoked: ``To say that truth is not out there is simply to say that where there are no sentences there is no truth, that sentences are elements of human languages, and that human languages are human creations.'' (Richard Rorty Contingency, irony and solidarity, 1989.)

Of course, a fastidious reader would require you give a pragmatic definition for `exist'. Mathematical objects, or properties, exist once their question is `evoked', but then they appear as completely invariant. The pattern is: we produce a change, and then we see that invariant phenomena appear, any one at any time would witness it. The pattern is the same when I bump into a stone: I produce a change (a step), and my toes find an invariant, that I call a stone. Any one at any time would find the same. Hence in some respect, the situation is the same, but I don't think we would say that the stone did not exist until it was `evoked' by bumping into it. However, we can certainly say something like that we were unaware of it.

There are similar small difficulties (I don't want to go into those set-theoretical-like paradoxes, whereby I would talk about Borges's Library of Babel which contains all the possible mathematical books, and pretend I evoked all the possible mathematics --that is not correct, since the exact invariants do not show yet): for instance, Ramanujam is famous for having `re-evoked' a lot of mathematics that others knew of already: obviously evocation does not `work' forever. It seems that we must make explicit the relativity of any mathematical object to those who know about it, and maintain it.

5. A path to improvement.

I think there is a way out of this difficulty. It requires, as is suggested by the above parallel, that we include explicitly in our axiomatic framework the processes of perception by which both stones and mathematics can be said to exist: They all appear as a sort of invariant, under particular conditions. This way of making explicit what it means to be `evoked' for mathematical objects is what I have attempted in my essay.

6. A Question.

You take some length to discuss, on one hand, games (typically board games like chess), and on the other hand, formal axiomatic systems.

For me, the two statements ``there are an infinite number of games we might invent'' and ``there is a potential infinity of FAS's (formal axiomatic systems)'' are simply equivalent. When I want to explain to someone not acquainted what a formal system is, I equate it to a board game like chess or go, and I deem it perfectly exact. (There is a sheet of paper in the Turing archive, where he has written down the rules of go strikingly in a formal system, axiomatic manner.)

Do you make any difference? Was that just rhetoric redundancy?

7. Two quotations.

That I feel fit particularly well here.

``And, moreover, we have found that where science has progressed the farthest, the mind has but regained from nature that which the mind has put into nature.

We have found a strange foot-print on the shores of the unknown. We have devised profound theories, one after another, to account for its origin. At last, we have succeeded in reconstructing the creature that made the foot-print. And Lo! it is our own.''

--Sir Arthur Stanley Eddington. Space, Time and Gravitation: An Outline of the General Relativity Theory. Cambridge University Press, 1920.

``A man sets out to draw the world. As the years go by, he peoples a space with images of provinces, kingdoms, mountains, bays, ships, islands, fishes, rooms, instruments, stars, horses, and individuals. A short time before he dies, he discovers that the patient labyrinth of lines traces the lineaments of his own face.''

―Jorge Luis Borges, The Aleph and other stories