How Mathematics Meets the World by Tim Maudlin

1. Dear Tim,

Your essay was for me the occasion to discover your theory of linear structures, and it was an enchantment. I think it won't be long till I buy your book.

Your text is particularly clear, too.

If I really like your style and your contributed linear structures, I have not really found an answer as to the effectiveness of mathematics, or, as you aptly rephrase it, how mathematics meet the world.

2. I refrained from commenting Wigner's article, because I see too many irrecoverable conceptual errors in it, for instance: ``elementary mathematics [...] entities which are directly suggested by the actual world''.

- The actual world suggests nothing.

I'd like to quote Ferán Adria, the famous Spanish chef; in answer to journalist saying that he was finding inspiration on the Costa Brava, he stood on the shore, staring at the sea and cliffs before a camera, and after a while said: ``No veo nada.'' There is nothing in clear in the landscape. Where is that inspiration?

- Wigner makes what I think is a huge mistake; implicitly, he suggests that there are things in the world that we can access in an absolute way. So, in the objects that I talk about, some are `for sure', and some are hypothetical. If you really try seriously to draw the border between the latter and the former, you come to understand that it does not exist, and that all perceptions are hypothetical. Only, some of them are usually, frequently confirmed, and we deem them solid ground. But, read Karen Blixen report of a earthquake in Out of Africa, and you will have a vivid case of a solid ground becoming moving and alive. Witness how people can be in a state of shock after a large earthquake; Haroun Tazieff used to say that their whole mental system is broken into pieces after experiencing the ``disappearance of space'', in the most concrete sense. Read Gregory Bateson's report of completely loosing ground after his experiments with sensorily puzzling effects at Adelbert Ames Jr. laboratory (in Steps to an ecology of mind). After that, read Richard Gregory's Perception as hypotheses (In The Oxford companion to the mind for instance). And read about Bach-y-Rita's experiment (guess where). You should convince yourself that the most fruitful representation is to consider uniformly all perceptions as hypotheses.

By the same token, you cannot talk about the `actual world', if you mean by that things you have a privileged, certain, absolute access to. All that you know about the world is relative to your perceptions, and is built through interaction with it, in action-perception loops: you impulse transformations into the world, and record the effects, and build pragmatically a whole world of hypotheses. You act, to abstract.

So, when you note after Wigner about the concepts of elementary geometry that ``those very concepts were developed via interaction with the world'', there is no solid way to circumscribe only them, because all your concepts, you eventually develop via interaction with the world. (You certainly agree: ``claims about [...] must be considered just as conjectural and fallible as all other physical claims.'')

(By the way, defining exactly what `elementary geometry' should be is not trivial, and finally not univocal because it poses a problem of foundation, and what we think is the historical, or natural genetic approach is not obviously better. Shall we just say that rays of light are our concept of a straight line? Shall we say that sets and groups are at the root? Shall we say that topos is a more elementary foundation?)

3. In the words of a common metaphor, Wigner sins by confusing the map for the territory.

(Korzybski for instance has repeated much this dictum.)

All what we do in our theories is language, and the world is not language.

The trains of spikes in your neuron chains are not labelled `vision', 'touch', `warmth', etc. These concepts are built afterwards. They do not exist substantially in the world, ready to be channeled to your perception. If you excite the retina by an electrical impulse, the subject reports a flash of light --the usual hypothesis for what comes from the retina. The most striking demonstration is in Bach-y-Rita experiment (refer to my essay for a pointer).

These facts should make clear that all we operate on is within the `maps' world. All we build in our maps is hypothetical constructs, that we try to fit to the interactions we engage with the territory that we know is out there.

So to be accurate, we must always maintain that we talk indirectly about the territory, through a representation. Any statement made directly about the territory, the world is problematic because it cancels this relativity to the representations of the subject. A statement about the territory itself, being absolute, must be completely right. Therefore, it cannot be scientific. It fails to satisfy Popper's criterion. If a statement fits Popper's criterion, then it is hypothetical.

It has nearly always been that progress has occurred when the operation of representation was made explicit and stated clearly. For instance, a correct definition of what a measure is, in mathematics, could not be derived until after there was a clear separation between the measurement, and the scale used to represent measurement, that is, a usable theory for the real numbers. I took an even simpler example in my essay, with the confusion between numbers-of something, and numbers `pure', and how until the late 19th century (!) not being able to separate clearly numbers from what they could be used for --counting objects or representing quantities in this case-- was impeding comprehension and progress in using them.

The conclusion is that to be able to address properly the question of the effectiveness of mathematics in representing the world, we have to include the stage of representation of the world in our framework. That is, we have to include the stage of perception and cognition in our representation, and that means including explicitly the cognitive subject.

4. I have the feeling that you have fallen in the same trap as Wigner, when you write:

``claims about the geometrical structure of physical space or space-time must be considered just as conjectural and fallible as all other physical claims.''

``A proper answer to this question must approach it from both directions: the direction of the mathematical language and the direction of the structure of the physical world being represented.''

In both cases, if I am not misinterpreting, by `physical space' and `physical world', you are not talking about the space or the world, as seen by physics, but about the world itself. You have no final word about the structure of the world. You have all latitude to propose hypothetical physical structures that prove fruitful for various uses.

5. Your title is particularly accurate and promising, because the key issue is exactly the meeting of mathematics and the world. I completely agree with it, and it is precisely what I have just said above: the issue is on perception. Perception is fundamentally a meeting, at any considered scale: Subject and event, sensory organ and object, sensory neuron and stimulus, etc.

If you think you talk about perception, and see no meeting, then you are certainly talking about things that take place during perception, but not the fact of perception itself, properly regarded.

Regards

Tim Maudlin Apr. 4, 2015: "Put two high-precision atomic clocks on the floor together Synchronize. Lift one up on a table. Wait a while. Return to the floor and compare synchronization. This has been done. The clocks go out of syntonization, and the amount out is a function of how long the one is up on the table. No redshift or light involved. Experiments at this precision have only been possible recently."

You abandoned this discussion after realizing that no such experiment has ever been done (no lifting and then returning to the floor). Gravitational time dilation has always been measured by measuring the gravitational redshift but the redshift actually confirms the variable speed of light predicted by Newton's emission theory of light:

Albert Einstein Institute: "One of the three classical tests for general relativity is the gravitational redshift of light or other forms of electromagnetic radiation. However, in contrast to the other two tests - the gravitational deflection of light and the relativistic perihelion shift -, you do not need general relativity to derive the correct prediction for the gravitational redshift. A combination of Newtonian gravity, a particle theory of light, and the weak equivalence principle (gravitating mass equals inertial mass) suffices. (...) The gravitational redshift was first measured on earth in 1960-65 by Pound, Rebka, and Snider at Harvard University..."

Pentcho Valev

"The basic scheme of actually comparing clocks has been going on for over three decades. These are not gravitational redshifts. Everyone knows the gravitational redshift is a weak test, but the tests moving and comparing clock readings are just different."

Such tests do not and cannot exist. As I have already said, these clocks are atomic oscillators that do not keep record of the time elapsed. So there is no point in lifting one of them and then bringing them together - there is nothing to compare. One can only measure the frequency difference (redshift) and ponder whether it is due to gravitational time dilation or variation of the speed of light with the gravitational potential.

Of course nobody is going to check which of us is telling the truth - you are right by definition. So... good luck!

Pentcho Valev

Initially you wrongly referred to this experiment:

http://www.nist.gov/public_affairs/releases/aluminum-atomic-

clock_092310.cfm

as an example of lifting one of the clocks and then bringing the two clocks together in order to compare them. Then you took refuge in the GPS system, and now you want to discuss the Hafele-Keating experiment which was not meant for measuring gravitational time dilation. Let us stop here.

Pentcho Valev

Dear Tim,

Thank you for your answer as it's a very interesting insight regarding how new theories come to be. I don't find it particularly odd that no one thought of your idea before, because most people don't feel the need to innovate as long as there exists a functional theory, all the more since the effort of creating a logically sound new approach is considerable. Also I'm sure your unique intuition has to do with this, because I think I understand that your approach allows for discreetness, not necessarily requiring infinitesimals.

Cheers,

Alma

Dear Tim Maudlin,

I will comment mainly on the first part of your essay. (The second part of the essay, in which you outline your new foundations for geometry, is interesting in it's own right, but it does not seem to address the original question as directly.)

One aspect that I liked is that you distinguish the surprise due to discovering a connection between seemingly distinct parts of mathematics from the wide applicability of mathematics to empirical sciences.

In the first part, you also think about how the world needs to be in order for the counting numbers to be relevant. You give mountains and cells as examples of concepts that are not (always) sharp enough (conceptually) for counting, and atoms as an example of a concept that is. In my view, however, the sharpness of our concepts is a matter of degree. The concept of an 'atom' is still a vague one, to some degree; hence the question of whether something is an atom or not -and whether to count something as an atom or not- may still be ambiguous in certain situations. For instance, one can imagine an electron and a proton: they will always 'feel' each other's electrical field. When, exactly, do they form a hydrogen atom? This question is also relevant in foundations of chemistry (with competing partition schemes in computational chemistry to identify 'atoms in molecules').

Viewed in this way, there are many degrees of freedom in applying mathematics to empirical findings - something which is rarely discussed in relation to the perceived effectiveness of mathematics.

Best wishes,

Sylvia Wenmackers - Essay Children of the Cosmos

Dear Tim, I very much enjoyed your essay and hope you can clarify one point for me. It seems that your argument suggests that there may be equivalent frameworks for describing physical systems (such as topology or linear structures) but that only one corresponds to physical reality. This suggests that we could be using the wrong mathematics to describe reality purely because distant branches of mathematics that seem unrelated are actually deeply related. How would we ever know then that we had the "right theory"?

Dear Tim :

Your essay is very interesting,

In my opinion, mathematics that reflects reality, remains a priority for the current science.The Bi-iterative calculation could be the right one, because it has all the characteristics of a pure mathematics.

"NUMBERS"

first are integers

second are geometric shapes

third are physical entities, with infinite degrees of freedom

fourth maintain always the aspect ratio

fifth are elastic ... ..

Sixth are compact, and unseverable by two egual part.

We take an example:

1 + 5 + 7 + 12 =

= 1 + 5 + 7 + (6 + 8 + 10)

= 1 + 5 + 7 + (3 + 4 + 5) + 8 +10)

= 1 + 5 + 7 + (3 +4 +5) + (1 + 6 - 9) + 10

= 1 + 5 + 7 + (3 +4 +5) + (1 + 6 - 9) + 10

= 1 + 5 + 7 + (3 +4 +5) + (1 + (3 + 4 + 5) - 9) + 10

= 13

=2197

Of course, we speek about a cubic equazion.

Recursive calculation, the Fibonacci series and Fractal are wrongly linear, because they consider the space as a bi-dimensional sheet.

The system's Bi-iterative calculation is different, it consider the space, multi-directional and multi-dimensional. In fact, exist only set and sub-set. with (1) we mean (1 * 1 * 1).

Because we live in a complex and varied reality, only a struture able to transform can describe it.

The human brain is divided into two halves, the left hemisphere is masculine, active, called "rational". The right hemisphere is feminine, passive called "irrational".

The most important thing is that these two opposing positions should coexist. (x + 1) and (x - 1) are two limits , i meam the linear structure and the non-linear structure are inseparable.

A system that can not stop,follow the arrow of time. Or rather, it coping itself, reflecting itself. So, we have (x + 1), + 1, + 1, + 1 ....

"The universe was born like that, one bit after another, and continues to do so even now".

(See ANNEX)"The teorem"

sincerly yours

BannouriAttachment #1: 1_1_Theorem_1.jpgAttachment #2: 1_1_Theorem_345.jpg

Tim,

A very interesting essay. I was skeptical at first but your explanation gradually resolved most of my reservations. Then I found just as it was getting to the climax with 'discrete Relativistic space-times' the last chapter was missing!

If you've read any of my recent essays (all finalists) you'll see various aspects of such discrete 'space-time' systems or 'fields' explored. I hope you may direct me where I may find your own last paragraph!

A few points;

1. I noticed your reply to Pentcho ref Hafele Keating above. If you'd read Hafele's NY lecture you wouldn't be so sure. He expressed his concerns that the data did NOT confirm theory as expected. It was only when the paper first didn't pass review that any such reference and data disappeared. The NY lecture text is also difficult to find (I can provide quotes if you wish). Nonetheless most still cite the work as the key excluder of many otherwise possibly valid theoretical approaches. Perhaps some gold plated 'confirmation bias'?

2. A question. Does your linear sequence model allow, say, expanding helical paths, and such paths which may form cascades on 'interaction requantizations' (via say atomic absorption and re-emission/'scattering', or perhaps 'decay' in particle terms) By 'cascades', consider a massive particle forming a shower through an atmosphere. The linear' structure would then have branches. Does your theory particularly exclude or support such models?

3. You say Wigners "..claim is indisputable', as commonly assumed. I'd like to challenge that. Lets say he had a friend 'd'Espagnolet' and they derived an 'inequality', later used by a chap called Bell, who cited a friend's odd red and green socks to prove its mathematical limits. All fine. But now let's give Dr Bertelmann a pair of reversible green socks with red linings. After their squash game they had to dress in the dark, so either foot could then have either colour! - giving a kind of 'stacked twin pair' probabilities, independent to each foot. Now we can have a perfectly physically feasible set of findings NOT constrained by Wigners 'amazingly effective' maths, or requiring 'quantum wierdness', or feet that can see in the dark and talk to each other! Some maths may then give the right bottom line but won't model the actual physical mechanism faithfully.

I propose the Wigner comment may then not prove indisputable, but that the situation he correctly identifies in the paragraph below applies - that there may be other truths. (A full description including complementarity referenced in my essay).

If we know that there are inconsistencies in current assumptions and theories is it right that we so consistently reject alternatives to doctrine? I just feel your own model may be on a track I recognize as being more widely consistent. Is "track" also consistent with your "linear", I've never considered 'lines' to exist any more than points!?

I do hope you'll also be able to read, consider and discuss my own essay, even after the score deadline.

Many thanks.

Peter

Tim Maudlin wrote on Apr. 17, 2015 @ 18:08 GMT: "I stopped the discussion because you do not know the situation with respect to either the predictions or tests of General Relativity. The basic scheme of actually comparing clocks has been going on for over three decades. These are not gravitational redshifts. Everyone knows the gravitational redshift is a weak test, but the tests moving and comparing clock readings are just different. Your insistence that they are not just demonstrates your mack of understanding of the situation."

I do understand the situation - you don't:

"A new paper co-authored by U.S. Energy Secretary Steven Chu measures the gravitational redshift, illustrated by the gravity-induced slowing of a clock and sometimes referred to as gravitational time dilation (though users of that term often conflate two separate phenomena), a measurement that jibes with Einstein and that is 10,000 times more precise than its predecessor."

"Einstein's relativity theory states a clock must tick faster at the top of a mountain than at its foot, due to the effects of gravity. "Our performance means that we can measure the gravitational shift when you raise the clock just two centimetres (0.78 inches) on the Earth's surface," said study co-author Jun Ye."

Pentcho Valev

"the discussion with Valev was not productive, and irrelevant to my paper in any case. The fact that he continues it despite being asked to stop (see below) is already indicative of a certain state of mind."

Thanks. Kind of you. I did stop the discussion but just found references (see below) disproving the following text of yours:

Tim Maudlin replied on Apr. 4, 2015 @ 23:49 GMT: "Put two high-precision atomic clocks on the floor together Synchronize. Lift one up on a table. Wait a while. Return to the floor and compare synchronization. This has been done. The clocks go out of syntonization, and the amount out is a function of how long the one is up on the table. No redshift or light involved. Experiments at this precision have only been possible recently."

Pentcho Valev

Dear Tim

I never learn topology, so I do not understand everyting. First I need to learn some background. But, it is known, that special relativity (according to Newtonian physics) defines causality. The now thing is also unsimultaneity. How can you connect your explanation with unsimultaneity?

I think that every new explanation from new aspect can tell a lot, so it seems to me that you have a good essay.

In my essay I speculated, that Pythagora theorem is consequence of kinetic energy conservation in ortogonal direction. Do you have opinion about this?

I thing also that Planck's dimensionless nature of space and time tells a lot ... Maybe still your approach fails to complete explanation.

My essay

Best regards

Janko Kokosar

Dear Dr. Maudlin,

Thank you. Your response is helpful. I tend to favor either the second or third possible position, probably inclining more towards the second of the three. I realize that the availability of a particular mathematical structure is not of itself an argument that an aspect of physical reality is one way rather than another. Still, it is important, I think, that we are not compelled to use a mathematical language which, if time is inherently asymmetrical, leaves that asymmetry out of the mathematical representation. It is good to know that there is an alternative language which would capture this important feature of the physical reality.

Best wishes,

Laurence Hitterdale