Wednesday, July 11, 2018

What's the purpose of working in the foundations of physics?

That’s me. Photo by George Musser.
Yes, I need a haircut.
[Several people asked me for a transcript of my intro speech that I gave yesterday in Utrecht at the 19th UK and European conference on foundations of physics. So here it is.]

Thank you very much for the invitation to this 19th UK and European conference on Foundations of physics.

The topic of this conference combines everything that I am interested in, and I have seen the organizers have done an awesome job lining up the program. From locality and non-locality to causality, the past hypothesis, determinism, indeterminism, and irreversibility, the arrow of time and presentism, symmetries, naturalness and finetuning, and, of course, everyone’s favorites: black holes and the multiverse.

This is sure to be a fun event. But working in the foundations of physics is not always easy.

When I write a grant proposal, inevitably I will get to the part in which I have to explain the purpose of my work. My first reaction to this is always: What’s the purpose of anything anyway?

My second thought is. Why do only scientists get this question? Why doesn’t anyone ask Gucci what’s the purpose of the Spring collection? Or Ed Sheeran what’s the purpose of singing about your ex-lover? Or Ronaldo what’s the purpose of running after a leather ball and trying to kick it into a net?

Well, you might say, the purpose is that people like to buy it, hear it, watch it. But what’s the purpose of that? Well, it makes their lives better. And what’s the purpose of that?

If you go down the rabbit hole, you find that whenever you ask for purpose you end up asking what’s the purpose of life. And to that, not even scientists have an answer.

Sometimes I therefore think maybe that’s why they ask us to explain the purpose of our work. Just to remind us that science doesn’t have answers to everything.

But then we all know that the purpose of the purpose section in a grant proposal is not to actually explain the purpose of what you do. It is to explain how your work contributes to what other people think its purpose should be. And that often means applications and new technology. It means something you can build, or sell, or put under the Christmas tree.

I am sure I am not the only one here who has struggled to explain the purpose of work in the foundations of physics. I therefore want to share with you an observation that I have made during more than a decade of public outreach: No one from the public ever asks this question. It comes from funding bodies and politicians exclusively.

Everyone else understands just fine what’s the purpose of trying to describe space and time and matter, and the laws they are governed by. The purpose is to understand. These laws describe our universe; they describe us. We want to know how they work.

Seeking this knowledge is the purpose of our work. And, if you collect it in a book, you can even put it under a Christmas tree.

So I think we should not be too apologetic about what we are doing. We are not the only ones who care about the questions we are trying to answer. A lot of people want to understand how the universe works. Because understanding makes their lives better. Whatever is the purpose of that.

But I must add that through my children I have rediscovered the joys of materialism. Kids these days have the most amazing toys. They have tablets that take videos – by voice control. They have toy helicopters – that actually fly. They have glittery slime that glows in the dark.

So, stuff is definitely fun. Let me say some words on applications of the foundations of physics.

In contrast to most people who work in the field – and probably most of you – I do not think that whatever new we will discover in the foundations will remain pure knowledge, detached from technology. The reason is that I believe we are missing something big about the way that quantum theory cooperates with space and time.

And if we solve this problem, it will lead to new insights about quantum mechanics, the theory behind all our fancy new electronic gadgets. I believe the impact will be substantial.

You don’t have to believe me on this.

I hope you will believe me, though, when I say that this conference gathers some of the brightest minds on the planet and tackles some of the biggest questions we know.

I wish all of you an interesting and successful meeting.

38 comments:

Rob van Son (Not a physicist, just an amateur) said...

"What's the purpose of working in the foundations of physics?"

I understand that those developing Quantum Mechanics were asked this question in the 1920s. Faraday was asked by Gladstone what the practical value of electricity would be. Faraday purportedly answered that it might one day be taxed.

We can only say that no physicist working on the foundations of physics ever knew the correct answer. Nor did anyone else know at the time. Also, we know that the value of electricity and quantum mechanics to the modern world would encompass almost all of the value produced in the economy. Without electricity and quantum mechanics, our world would be much, much poorer than it is now.

So, empirically, we know that this question cannot be answered now in any meaningful manner. On the other hand, history has shown us that the actual value of this work to the people of the future will surpass out wildest fantasies.

Rogier Brussee said...

When Gladstone, then British Chancellor of the Exchequer (minister of finance) asked Faraday of the practical value of electricity he purportedly replied "One day sir, you may tax it." It is most likely an invented quotation, as there are no contemporaneous records, but it is a nice quote anyway.

Of course these day, such a question comes on an attached form that want you to estimate the number of jobs created in two years.....

So here is a great quote from Feynman: "“Science is like sex: sometimes something useful comes out, but that is not the reason we are doing it. ”

sean s. said...

I'd be interested in the questions or comments your remarks may have given rise to.

sean s.

Dr. Decay said...

Sabine
"And if we solve this problem, it will lead to new insights about quantum mechanics, the theory behind all our fancy new electronic gadgets. I believe the impact will be substantial."

I believe you. On the other hand, concerning "What is the purpose of …?", you write:

"No one from the public ever asks this question. It comes from funding bodies and politicians exclusively."

Really? I'm much less experienced in talking to the public than you are, but once a year or so, I give public tours of my lab. I don't work on "foundations", but it's pretty basic research. I am occasionally asked: "Are there any applications?" That's the same question. No?

OK, perhaps a lab full of lasers and vacuum systems attracts a different kind of visitor, or stimulates different questions than a talk about say, black holes and the multiverse. But I'm pretty sure the people asking weren't funding agents or politicians.

I love your blog.

Space Time said...

I think a better question, or a way to look at the title question, is: What are the problems people in the foundation of physics work on and why are these problems interesting?

Uncle Al said...

A beam of volatile, identical, small, rigid, internally multiply-connected, homochiral molecules traverses a diffraction grating (matter, Markus Arndt; photons Anton Zeilinger).

… 1) The expected pattern emerges. Molecules must racemize by Hund's paradox. 100% left or right shoes entered. A mixture must exit. Chemistry is falsified. Calculated energy (temperature) needed to break the bonds to do this is unavailable (even at the Boltzmann distribution’s tail).

… 2) No pattern emerges. The beam remains homochiral. Quantum mechanics is falsified.

… 3) Somewhere in-between. Everybody furiously publishes.

Are physics' postulates empirically exactly true? Baryogenesis, non-classical gravitations, standard model plus SUSY, dark matter, neutrinoless double beta-decay say “no.” Stop philosophizing. Look.

Matthew Rapaport said...

Faraday's answer brilliant and prophetic! There is your perennial answer to the "purpose question" Dr. H! "What I learn will one day return a profit" will always get you that grant! :)

andrew said...

One practical benefit of a more fundamental theory than the SM plus GR would be that a more fundamental theory might have fewer experimentally measured constants and elucidate exact mathematical relationships between them.

For example, suppose that your more fundamental theory allowed you to derive all fundamental particle masses from the Higgs vev and all SM coupling constants from the U(1) coupling constant of the Standard Model, both of which are known to great precision. This would once again leave us in a situation where theory predictions were mostly much more precise than experimental measurements for the affected physical constants.

One aphorism regarding how to evaluate the value of a theory is the extent to which it provides information for free, without necessitating further work in measurement. By that measure, most of the SM is a pretty deficient theory, since lots of its theoretical calculations are no better than the accumulated experimental database of measurements. But, a more fundamental theory could increase the value of the more fully understood SM dramatically and quickly.

Kevin Van Horn said...

One could similarly have asked logicians in the late 19th and early 20th Century, "What is the purpose of working in the foundations of logic and mathematics?" It seems like highly esoteric philosophy with no practical applications.

But in the course of understanding what it means for there to be an "effective procedure" for listing a set of axioms, they invented the notion of computability, and the idea of a programmable computer (universal Turing machine), leading to a new industrial revolution.

Lawrence Crowell said...

For all we know we may be just an expanded version of some protistan species that has it figured out, to the extent a paramecium or such a species can figure, that the universe is based around some molecules it can receive from its environment. Well obviously we are a lot more than that, and we do know about our solar system. I bet there are few planets with beings who know about there own solar system. I bet there are fewer places where there are remnants of a spacecraft that carried such beings to another planetary body. Think Tranquility Base. We know about atoms, particles, the Higgs field, gravitation and a bunch of other stuff. This could though represent a near infinitesimal slice out of the totality of existence.

It is not clear that this means anything at all. We so far have no clue that the universe has any meaning other than it exists. The universe for all we know could just be a huge jestoresque fluke, and we are a fluke within a fluke along for the ride. So is there any great existential purpose to our knowing about the universe or studying foundations? Ideas about purpose to existence tend to involve religion. Accounts of God are obscure and unsubstantiated, and the last of these accounts had Him getting a Jewish girl in trouble, getting Himself born in the act, playing suicide by cop and then getting Himself back to life to rise into heaven. Since then, except for the occasional piece of toast that looks like His face He has been really quiet. So those accounts for a purpose to our existence or doing anything at all are unreliable.

When if comes to posterity or the immediate future, that too looks dicey. We humans are a terminator species, or the ultimate invasive species, who are demolishing the rest of life here and converting everything we can into trash. Such a terminator species is likely to be a self-exterminating species, and the future time table for our collective suicide appears to be not that long.

The only reason to look at the foundations of physics or other such things, that can be the creative arts or other studies, is because we enjoy doing them. There really is no other reason for it, even if with our demise this has not meaning or purpose.

I tell you, we are here on Earth to fart around, and don't let anybody tell you different. Kurt Vonnegut

Ajit R. Jadhav said...

``Let me say some words on applications of the foundations of physics.''

Oh!

Best,

--Ajit

qsa said...

I think the importance of foundation ( a very good one, nearing ontology) Should decide that there is either God or there isn't. And That should change Human brain thought process and his actions in dramatic way , basically changing human history.

Bill said...

With all due respect, folks in the field of the physical sciences who ask that question ought to choose a different field.

Enrico said...

What is the purpose of fundamental research in physics? Had our ancestors not done it, we would still be living in the Stone Age. The ancient Egyptians knew and used the six simple machines. They did not call it mechanics but they built the Great Pyramid with mechanics. All of engineering and technology are built from fundamental principles. A few crazy nerds discovered some fundamental principles and the rest of humanity benefits from it for as long as humans exist.

Einstein was invited to run for President of Israel. He was the most popular Jew at that time. He politely declined and said, “Politics is for the moment, an equation is for eternity.”

Bhupinder Singh Anand said...

Dear Sabine,

Appropriate; and very well expressed.

Bhup

Phillip Helbig said...

"The reason is that I believe we are missing something big about the way that quantum theory cooperates with space and time."

What do you think is the most promising way forward? Loop quantum gravity? Something else?

John Fredsted said...

Once again a pleasure to read, thanks.

Bhupinder Singh Anand said...

Dr. Decay
 
That's the same question. No?
 
Unlikely.
 
Visitors to your lab were probably curious about what you did, and why you did it, for broadening their own perspectives.
 
I doubt whether funding bodies and politicians share the same curiosity.

Reminds me of the moment in 1985 when I was investigating the structure underlying the generation of primes. I had coded the program into the assembly language of a ZX80+ Spectrum, and hooked it up to a printer with output simultaneously on the family TV, which continuously flashed a bold notice 'Do not switch off'.

After a few days of being denied any home entertainment, my father---an ex-air force officer and founding partner of our small family business, who had been urging me to look at the emerging computer technology commercially--- asked in all seriousness: "Will this make any money?"

I think it was my involuntary burst of laughter which perhaps reassured him that what I was pursuing---and had been pursuing since my college days---had an intangible value that---to his credit---merited his continuing indulgence.

Regards,
 
Bhup

Tam Hunt said...

Nice piece. Intellectual curiosity, and satisfaction of that curiosity, both personally and collectively, are fine motivations in themselves. Any "real world" application beyond that is icing on the cake but shouldn't be the raison d'etre of pure research. I think our modern societies are sophisticated enough to understand that and we have collectively agreed that pure curiosity is worth funding.

David Bailey said...

Sabine,

I think you have hit the mark with this post, because gadgets can be good or bad (e.g. nuclear weapons), but they all provide a reality check for scientific theory.

I think it is striking how the gadget test suggests a problem going decades back. How many gadgets have emerged from quarks, or any of the particles with lifetimes shorter than (say) 10^(-20) secs.

I am not a physicist, and my quantum chemistry is extremely rusty by now, but I think you have put your finger on something I have felt for a long time!

marten said...

I think that working on the foundations of physics, the nuts and bolts so to speak, is a very good purpose of working in the foundations of physics.

David Schroeder said...

The research in Quantum Gravity that Sabine, and others, are pursuing may someday have a major, practical impact on our civilization. Offhand one can imagine that the fruits of such intellectual labors might lead to a future technology allowing manipulation of space-time in a very substantial manner. That, in turn, could revolutionize the aerospace field, which currently is entirely based on ejecting hot gasses or high velocity air rearward to achieve forward motion. While, admittedly, this is pie-in-the-sky speculation there's quite a history of pie-shaped objects flying through our airspace, without obvious means of propulsion, and not just on TV where contestants are plastered with lemon meringue filling.

Tam Hunt said...

@Philip and Sabine, my feeling is that we need to quantize space and time and reinterpret the Lorentz transformations (SR) in a way that recognizes that space and time are in fact fundamentally different features of our universe. Whereas movement in space is fundamentally symmetric movement in time is fundamentally asymmetric. This requires sacrificing a sacred cow or two but it's very likely the cost of real progress on quantum gravity.

Koenraad Van Spaendonck said...

"The reason is that I believe we are missing something big about the way that quantum theory cooperates with space and time."

My proposition :

Get rid of infinities in every corner of a new theory, along with any and all infinitesimal calculus.

Unify the propagation of light with a discrete gravitational field.

Hypothesize on how relativistic time is a measure of energy density.

Hypothesize on how space density equals energy density.

Implement everywhere that energy comes in quanta.

Find a causal principle for gravity at the heart of it.

Aim for a generalisation of GR while respecting GR effects, but not necessarily current interpretations.

Model the above into one geometry, and investigate the consequences, the validity and the prediction(s) with experiments.


Best, Koenraad

Peter Shor said...

From the beginning of a 1977 book review on Donald Monk's book Mathematical Logic.

"On the banks of the Rhine a beautiful castle had been standing for centuries. In the cellar of the castle an intricate network of webbing had been constructed by the industrious spiders who lived there. One day a great wind sprang up and destroyed the webs. Frantically the spiders worked to repair the damage. For you see, they thought it was their webbing that was holding up the castle."

Thirty years ago, this charming tale was regularly told to students specializing in logic, to leave them in no doubt as to how their field was regarded by their fellow mathematicians. Even a casual examination of Monk's book makes it apparent that nowadays the "spiders" are to be found all over the castle.

So the foundations of mathematics turned out to be not so useless after all. One can hope that the foundations of physics will be similarly useful.

Arun said...

Faraday and what use is a baby:
https://www.jstor.org/stable/986790?seq=1#page_scan_tab_contents

Dirk Freyling said...

…My first reaction to this is always: What’s the purpose of anything anyway?
Why do only scientists get this question?...


Theories are developed by their authors with regard to already known or suspected phenomena. That these phenomena can then be derived from the theories, although not always very convincing, is not very surprising.
Theoretical models always lag behind the (measurable) reality. In this context, it is amazing how many "interested" people and "science professionals" believe that quantum field theories are innovative theoretical concepts that generate practical applications.

On closer inspection, the hundreds of theoretical papers based on Quantum field theories published each month on arXiv.org do not present any significant new findings. They are just mathematical possibilities.

If physicists believe that basic physical equations preceded physical phenomena, or that reality is made up of mathematical formulas, they make an analog mistake like Plato, who derived reality from concepts rather than concepts from reality.

Don Foster said...

Feynman wrote about spending hours watching ants and performing little experiments by disrupting their linear paths. Others have noted that this experience likely influenced him in developing his path integral formulation. Perhaps you never know when innate curiosity will be rewarded. What's to be done about it in any case?

akidbelle said...

Hi Sabine,

you write: "In contrast to most people who work in the field – and probably most of you – I do not think that whatever new we will discover in the foundations will remain pure knowledge, detached from technology."

At last!.. Thanks. I agree with this statement, and I believe the next leap is as big as going from Newton to QM.

But I guess many will politely answer "maybe, but not in this century" or the likes. Right?

Best
J.

Bhupinder Singh Anand said...

Dear Dirk,

If physicists believe that basic physical equations preceded physical phenomena, or that reality is made up of mathematical formulas, they make an analog mistake like Plato, who derived reality from concepts rather than concepts from reality.

Succint, and to the point. Expressed less succinctly:

Seems to me that all scientists could profit from cognitive scientists George Lakoff and Rafael Nunez’s book, ‘Where Mathematics Comes From’.

The authors cogently argue that what we express in our mathematical languages are essentially conceptual metaphors which are grounded in our sensory perceptions; not only of the reality that we directly experience, but also of the symbolic representations of such reality that is experienced by other intelligences---whether human or mechanistic.

Moreover, the language most naturally preferred by a human intelligence is that of a set theory such as ZFC, since it adequately expresses---in what is treated as a subjectively 'unambiguous' symbolism---our directly experienced spatial and temporal sensory perceptions.

If so, it follows that the theory being considered at any instant by an individual scientist is an attempt to subjectively conceptualise a, holistic, perspective of the individuals internal ‘universe of conceptual metaphors’; one which, moreover, is seen as satisfying some individually decided yardstick of what may be termed as ‘holistic’.

Such a perspective, as you have pointed out, should not be conflated with the external reality in which an individual’s personal experiences (such as that, for instance, of free will) are grounded.

Since, under Lakoff and Nunez's thesis, the scientific method would mean isolating only those elements of such a theory that are common to the individual, internal, ‘universes of conceptual metaphors’ of a community of individuals who express their conceptual metaphors in a common language (symbolism), two question arises:

(1) Is there a language that will allow us to isolate and unambiguously communicate such common elements?

The answer was, of course, provided by Turing, Church and Goedel in the 1930’s; each of them separately identified the language in which---as Mandelbrot demonstrated subsequently in the late 1970’s---natural phenomena is best expressed.

This is the algorithmically computable language in which mechanical intelligences (such as, for instance, those guiding the autonomous operations of a space craft) record and communicate their ‘sensor-based’ perceptions of their external reality.

As shown in this paper, the language of precise expression and unambiguous communication for a human intelligence is the categorical first-order Peano Arithmetic PA, whose ‘propositions’ are true under a unique, finitary, interpretation over the natural numbers if, and only if, the ‘propositions’ are evidenced as algorithmically computable ‘truths’ by the finitary convention regarding 'truth/falsity' built into a mechanical intelligence.

(2) Can the set theory ZFC---which is the language of theoretical scientists---also admit the isolation and unambiguous communication of such common elements?

The answer here is, as obviously, in the negative; since the axiom of infinity of ZFC has no demonstrably-consistent interpretation under which the propositions of ZFC could be verified as ‘true’ by any ‘objective’ convention.

Ergo, any physical theory which cannot be communicated in a categorical mathematical language, such as PA, has elements that refer to---essentially ephemeral---conceptual metaphors which exist at that instant only in the mind of the individual interpreting the theory at that moment.

Moreover, any such theory---no matter how insightful---cannot serve as a foundation for, an unambiguous perspective and understanding of that part of the universe accessible to our sensory perceptions.

Kind regards,

Bhup

Unknown said...

Years ago, I was teaching undergratuate "Stars and the Universe," and in the first lecture I was flummoxed when a student asked "why do we care about what's out there in the universe - it doesn't matter to us, does it?'

Rob van Son (Not a physicist, just an amateur) said...

"why do we care about what's out there in the universe - it doesn't matter to us, does it?"

That is not limited to physics. An American linguist once told me she had the same question asked about languages. As every human speaks, or should speak, English, learning about foreign languages was a waste of time.

Physicists should not feel guilty. It is not their craft that is useless, it is the mind of those that pose the question that is unable to see benefits until they hit them in the face. These are the people that told us that the iPhone was a fad, as were the WWW, email, personal computers, lasers, and the telephone.

When someone asks me why someone would study Astrophysics or Religion I always respond that it is fun and can make you happy. When they ask why society should pay for it, the aswer is that the ROI of basic research has historically been estimated in the 20-30% per year range. The economic value of the WWW alone is in the order of 5% of GDP in the major economies. That value alone dwarves everything that has been spend on CERN or even particle physics.

So, science, for fun and profit.

David Bailey said...

Unknown wrote,

"Years ago, I was teaching undergratuate "Stars and the Universe," and in the first lecture I was flummoxed when a student asked "why do we care about what's out there in the universe - it doesn't matter to us, does it?'"

For much of my life, I too would have been flummoxed by that remark, but now that we have the abstractions of string theory and an LHC that gave us just one new particle that lasts for about 10^(-25) sec, multiple universes, etc, I'm not so sure. Maybe claiming to know about the creation of the universe(s), how matter came into existence, how the universe(s) will end, etc is just hubris. The number of assumptions and approximations that I am sure have gone into all this, gives me a sour sense that the great enterprise of science may be curdling.

Wes Hansen said...

Roy Wagner has some nice critiques (constructive) of the Lakoff/Nunez work:

A historically and philosophically informed approach to mathematical metaphors;

Infinity Metaphors, Idealism, and the
Applicability of Mathematics
.

Isn't work in the foundations really a search for hidden assumptions, the revealing of which leads to greater clarity and possibly greater rigor - if not new foundations entirely?

Matthew Rapaport said...

Both good papers Wes Hanson, thanks

mls said...

Bhup,

First-order Peano arithmetic is not a categorical theory. This had been first shown by Skolem. Modern proofs invoke the compactness theorem of first-order logic on a language expansion consisting of a single constant. For each finite sequence of numerals, the constant is held to be greater than those numerals. Because this is satisfiable for every finite collection of numerals, it is satisfiable for the infinite collection of the standard model. Thus, there is a model of Peano arithmetic containing a number which is unreachable from 0 by unit succession.

On another point, to declare the membership relation of Zermelo-Fraenkel set theory to be representative of spatial and temporal intuition is a fabrication of history. I am certain you are only repeating what others have said. Zermelo-Fraenkel set theory is a product of logicism. Its notion of empty set as numerical zero comes from Frege who gave a compelling argument against temporal intuition as a basis for arithmetic. One could -- as Lawvere and McClarty have -- hold that the views of Cantor and Frege differed significantly. Perhaps Cantor's topological leanings justify treating membership more like geometric incidence. But Zermelo chose to follow Frege on certain matters. This, at least, is how Lawvere and McClarty interpret the history. Not only is it inappropriate to portray Zermelo-Fraenkel set theory in this way, but, the community of analytical philosophers who are committed to this account adamantly reject "intuition" as a basis for mathematics.

There is a real problem here because "core mathematicians" choose not to participate in foundations while accepting results with complicated presuppositions that may not reflect their views.

For what this is worth, understanding Zermelo-Fraenkel set theory is further complicated by formalism. Skolem also showed that its first-order presentation is not categorical.

What I am trying to convey is that one cannot simply ignore what has been done in the foundations of mathematics to satisfy what one might desire. There is an oblgation to address arguments underlying received views. Moreover,there is an obligation to produce independent arguments that must withstand criticism.

I do not necessarily view homotopy type theory as a legitimate foundational paradigm. But, it is exactly the kind of approach one must take. And, it chooses Lawvere's category-theoretic account of sets rather than making unjustifiable claims about Zermelo-Fraenkel set theory.

mls said...

Wes Hansen,

It would be nice if foundations were that simple. Consider, for a moment, how one is to understand truth. Naively, we believe truth in terms of a correspondence theory. If, however, we ground our mathematics in the subjective experience of individuals, the most that one can hope for is a coherence theory of truth. This is important because mathematicians communicate with proofs, and, the syntactic transformations which comprise a proof are intended to preserve truth. This is how a rule of detachment differs from a rule of inference.

The recieved view is that of first-order logic with a correspondence theory of truth based upon Tarski's work. Even this is not entirely clear. First-order logic actually invokes an analytic conception of truth, whereas Tarski introduced a semantic conception of truth. In a semantic conception of truth, non-denoting terms force atomic propositions containg them to be false. So, one can actually have "x=x" to be false. In first-order logic this is always true.

Many scientists and "core mathematicians" want to ignore issues like this. And, they will seek authors who explain issues in terms they find appealing. But, actually finding authors who make their statements rigorous is another matter.

One of the papers you mention speaks of points in Eucldean space. It presents points as corresponding with Cantor's nested closed set theorem with vanishing diameters. There is no mention of "closed" in the account, although this is very important. It cannot be proven for open sets (in fairness, "closed" might be assumed in the definition of "disc".

Someone more familiar with formalist foundations would point you to Hilbert's book, "Foundations of Geometry". You will find no such account of "point". Instead, one has "point" as an undefined language primitive. Hilbert's book is so appreciated in foundations because it eliminates "intuitive" notions such as "drawing" a line.

When you do take familiar geometric ideas into account, one might end up trying to argue for hyperreals as a more accurate account of geometric intuition. But, this leads to second-order logics. In turn, this leads to objections from those who view first-order logic as mathematical logic.

Foundations is hard. And,as with scientists, the people who wotk in the area have diverse views and motivations. Not all of them are compatible with how we would like to think about the subject.

Unknown said...

In my opinion nature as a whole is the true (relational) fundament of physics. The rest of our knowledge of physics is incomplete abstract knowledge.