Clarifying the foundations : maths, physics and more.

Introduction to the site settheory.net by Sylvain Poirier

I am a math PhD from France, and develop the site settheory.net to clarify the foundations of math and physics, by self-contained undergraduate level courses combining depth, generality, rigor and concision, involving many simplified but accurate presentations of concepts usually considered graduate level. I do it for free outside any institution.

Leaving academic "research" behind

I was always interested in mathematics, particularly algebra and geometry (especially in my youth). More generally, I am interested in global, foundational issues in diverse fields and the search for perfect theoretical solutions. During my teenage, I was very unsatisfied with the official teaching, which was an intellectual desert for me. I managed to learn Special Relativity and then express General Relativity by my own research. My dream was to become a particle physicist, as I naturally guessed that the fundamental laws of physics were rich of some of the most wonderful mathematical theories. But things turned out very differently, for many reasons.

A first trouble was the low level of the teaching curriculum I went through, which I so direly needed to escape in my free time to explore theoretical physics far away from it, an exploration I thus did alone outside any academic guidance. But people around almost objected to this exploration, insisting that it wasn't time for me to explore the high skies of theoretical physics because I first needed to more fully assimilate the basic concepts on which higher theories can be built. In fact there is some truth in this idea, but in a very different way. It is not that I shouldn't have gone to theoretical physics at the beginning because I needed the help of schools (which I hated) to first teach me the right basis, but that I'd rather not continue to further theoretical physics at the end because schools (precisely: undergraduate curricula) actually don't provide the right basis for this, so that my help is needed to explain how to do it (but away from the system, which isn't welcoming any such proposition of progress).
As in high school I was isolated in lack of hints for going further (to quantum physics...), I spent time re-thinking and finding ways to progressively clarify the same theories (general relativity and electromagnetism). Still it wasn't enough, so I also thought about metaphysics (absurdity of the AI thesis which reduces the mind to a mathematical object), and economics (foundations of money and its stabilization problem; what problems occur in the world and which logical structures could ideally resolve them).

At one point of my studies (in ENS Ulm), someone warned me that particle physics was actually a terrible field to work in. Following that advice I didn't continue far enough to fully verify this claim by myself, however I see indirect reasons for this to be true. One is the mess of divergences that the fundamental equations of quantum field theory are plagued of, making it unclear whether any possibility of a rigorous mathematical approach should reasonably be expected at all. Another is, that many of the brightest minds from around the world are already working on the problem; so, what could just one more bright mind add to that, unless he'd have both incredible lucks of being somehow "the brightest", and that Nature would have chosen the ultimate laws of physics (or something amazing in them) exactly appropriate to be only discovered by the brightest physicist but not by others without him ? Wouldn't it be both more polite and cheaper in efforts, in case an amazing new discovery was just ahead of us, to leave the honor of making it to someone else (who would need it, for some reason I don't fully grasp but...), and simply later learn about it if really interesting ? The golden age of physics research, quickly grabbing lots of low hanging opportunities of discoveries, which Nature had available for us, seems to be past. After just a few decades of rapid advances (that is the blink of an eye compared to the long history of life on Earth), wonderfully explaining most of the physical phenomena that could be experimented, why did this quest have to suddenly stall without being complete ? God knows ; anyway we must cope with it.

Generally, I was quite disappointed with the academic environment of mathematical research I experienced during my PhD (in algebraic topology : Vassiliev invariants constructed from the perturbative expansion of Chern-Simons topological quantum field theory, a topic I abandoned since then). The main group activities (what makes a life in official positions and labs differ from just staying home doing free personal research) were regular seminars where people reported their boring ideas to others who anyway don't care and cannot use in their own research (as they don't work in the same topic) but who anyway feel obliged to come and pretend listening to just for the sake of politeness (I was once scolded for not coming, since "being polite" in this way visibly mattered more than any meaningfulness of time spent). Indeed, why would the ideas of whoever happens to work in the same lab be worthier hearing than those of any other people working elsewhere on Earth ? that is something I couldn't figure out. I also several times heard of stuff like HOMFLY having multiple independent co-discoverers. Why bother researching and discovering something, just to end up being one of several independent discoverers of the same thing ? Even if institutions were okay to pay me for this, it didn't feel to me like the best way of giving sense to my life.

Other factors drove me away from academic research, which I left for good after one year teaching as assistant professor of mathematics at Reunion university. It would still take me quite a long further study to catch up existing works until I could produce valuable new results myself in great fields such as particle physics (I had the chance to manage my PhD by picking a topic with not so great value to me, but wouldn't see what more to do there), while I was already exhausted by the wasteful academic path I previously followed with disgust as it wasn't done the way I believed to be needed. I also felt that the kind of popular hard unsolved problems which scientists usually work on, aren't often the most valuable ones:

The popularity of a topic cannot define its real value : the correlation may rather be negative, as popularity indicates that its possible low hanging fruits have been already picked;
I once read a report of a result of complexity theory that "hard theorems are useless" but lost the reference; anyway the point is that things appeared this way to me in practice.

Generally, it seems too many people focus their works on tackling some of the hardest questions, a bit like Olympic champions training their strength for the purpose of demonstrating how strong they are, but are not so good at asking the right questions (and I do not see academic philosophers better at this either despite their claims). A typical example is Bitcoin: a wastefully sophisticated answer to the wrong questions about how a good online currency should work. Hardly any of its proponents seems to have seriously wondered for instance how the value of a currency could be stabilized, or even understood the fundamental importance of this question.

Also, institutions have fundamental flaws such as the Peter principle (workers are tendentially raised to their "level of incompetence"). As I want to make an optimal use of my work for the world, I rather search for the right questions or tasks, where I can produce the most valuable works not because I'd be the most clever, but rather because these are crucially important questions on which, strangely, no other good thinker seems to be working yet. But such a radical form of originality, of choosing a research topic or other working direction whose potential value is ignored by the rest of the world, also makes unlikely to find any job open for this in any institution (since job openings are set by administrators more often looking for security than originality, while, logically, widely unexpected discoveries are hard to expect). I explained more aspects of my reasons for leaving the academic system and why it was a bad idea to join it in the first place, in my video "Why learn physics by yourself"

From physics to mathematical logic

I was initiated to mathematical logic with amazement during my graduate studies (Magistère, ENS Ulm). It took me some maturation time until I could be productive there. Like in physics, I focused on simpler aspects, for which I could be directly useful: clarifying the basics (so, the foundation of the foundation). Things which many students have to go through but in need of clarification (to be done once and for all), usually neglected by other researchers who focus on more complex, high level issues with their hard open problems. But, as long as researchers only care to do research for the sake of "doing research", their productions are likely to stay desperately hidden in their ivory tower, useless to anyone who isn't yet already another researcher in the same field, having wasted many years of their life following traditional curricula followed by the other lengthy wasteful initiation path to that particular field of specialization.

Among the people to whom an initiation to mathematical logic could benefit, but who currently cannot afford it because of the wasteful complexity of its teaching path, there are... physicists who have no time for this long path because they already wasted too many years of their life with their own messy teaching curriculum. But, one of the things that made them waste so much time in their own curriculum, is... their weakness in mathematical logic. But what could be the use of mathematical logic in physics, you may ask ? It is of course that the laws of physics are deeply mathematical, as reported by many great minds. Between the first guesses by Plato and Pythagoras, and some recent reports such as Wigner's "unreasonable efficiency of mathematics" (confirmed by Hamming in 1980), a remarkable short formulation is the one by Galileo:

"Philosophy [i.e. physics] is written in this grand book — I mean the universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth."

But, once confirmed that physics is indeed written in the language of mathematics, we still need to specify what this language of mathematics actually is, for better clarifying our understanding of physics. Precisely, I found 3 interesting connections between physics and mathematical logic. A first one is a lesson which logicians have ready to teach to physicists. The second is an interesting formalism of mathematical physics usually ignored by logicians but in need of logical clarification for teaching. A third (but debatable) one is a possible lesson from logic for philosophers of physics.

First is the simple issue of what a theory is. Physics has its list of theories; mathematical logic has precise general concepts of how a mathematical theory can be formed (as can be defined by either first-order logic or second-order logic, but their differences do not matter for physics), but strangely, physics teachers never seemed to care whether any match between these should be made and how. And which interesting lesson can logic have to say to physics teachers about what a theory is ? That each theory comes with its own language: the language in which it is expressed, made of a list of names of the structures of the system which the theory describes. Then, we may introduce and study the automorphism group of the system, made of all transformations which preserve these structures.
But physicists took completely different habits: instead of this, they just express any of their theories in always the same language (the only one they know, as if it was the only possible language of mathematics): the language of packs of real numbers and packs of operations. Then, among their packs of operations they choose some specific ones: some transformation groups. And only at the end of this, they investigate which of the many possible operations have a special property : the property of being invariant by the given transformations (which they may name by some more sophisticated adjectives such as "covariant" to make it look like an amazingly intelligent quality, no matter if students cannot decipher what it may really mean). But did physicists consider that in logic, these invariant or "covariant" structures are those which come first and are actually the only structures which exist in the language of the intended theory, in terms of which the theory naturally ought to be expressed ? Did they ever consider that they might have just wasted a lot of time and obscured the understanding of the theory in the students minds, every time they expressed something in formulas formed of variables and operations which don't already belong to this short list of invariant stuff ? Do I need to explain why such a way of "mathematically expressing theories" can be a terrible one, making the physics teachings much messier and harder to follow than actually necessary ? (I was recently censored from physicsforums.com just for the sin of defending this view, that there exists such a thing as a mathematical conceptualization, and that a explanation's quality of "being mathematical" cannot be reduced to how numerically accurate it is ! Namely, I was denied the right to reply to an accusation that my approach was "without using any math" just because it wasn't purely computational)

Second, comes the issue of how formal expressions are structured. Many logic specialists, in a desperate try to kill boredom and give themselves some jobs, spend a lot of time inventing and investigating their own alternative logical frameworks, eventually involving some new extravagant ways of putting symbols together to form formulas, regardless that nobody beyond them has any chance of ever making use of such formalisms anyway. But hardly any of them seems to have heard of the following facts, actually well-known by anyone who studied graduate physics, just because... most logic specialists never went to learn graduate physics themselves:

A large part of the concepts and formulas throughout theoretical physics (from relativistic mechanics, electromagnetism and General Relativity to basic quantum physics and quantum field theory), is actually expressed in a common mathematical formalism: that of tensors which looks quite different from all the "normal" style of mathematical formulas (the only one known by undergraduate math students) described by classical logic.
What is remarkable there is that each (monomial) tensorial expression forms a graph made of occurrences of symbols linked together in ways respecting their types (reflecting the spaces to which arguments respectively belong), similarly to how, in classical logic, symbols are normally linked together (each operation symbol to its arguments) to form expressions, but without any definite hierarchical order of sub-expressions from a root to branches.
The strangeness of this formalism in which physicists are already expressing much of their works, still makes it a big trouble for them to properly explain what it all means. Their usual initiation courses to the topic remain so messy and hard to follow, that this "difficulty" forms one of the main obstacles against introducing it, and with it some "more serious" physics, into undergraduate physics curricula.

Third, is the philosophical understanding of the nature of time. General Relativity only involves time as a geometric dimension, thus without any philosophical feature of time such as its flow and its orientation. Then quantum physics and statistical physics do involve a time orientation and flow (with measurement and entropy creation), however these seem to come "from nowhere" and are not really accounted for by any theory of physics. Mathematical logic, on the other hand, explicitly provides some kinds of time orientation interestingly similar to intuitions of the flow of time and free will (though usually not pointed out) : complexity theory, randomness (Chaitin), halting problem, truth undefinability and incompleteness, transfinite induction... I already introduced the main aspects of this flow of the time of mathematics in the philosophical pages at the end of my Part 1 on mathematical foundations. I think these similarities could be worth entering the philosophical debate, and I already formulated my propositions for this in my metaphysics text A mind/mathematics dualistic foundation of physical reality.

Mathematical foundations

To sum up some of the main points of my contributions in the foundations of mathematics:

While I agree that ZF(C) is one of the best references for research works on set theory, especially for relative consistency results such as that of CH, I find it not the best choice of basis to start mathematics: neither for "practical mathematics" for undergraduates, nor even as properly self-explaining of why it is indeed a good reference for high-level consistency results.
For a better start of mathematics I propose a new formalization of set theory, introduced in parallel with the rudiments of model theory. There I admit as basic objects not only sets but also pure elements and functions. Oriented pairs and other tuples are more cleanly defined as functions.
Several axioms of sets existence come as particular cases of a single principle, a sort of "any class behaving like a set is a set", namely : the classes in which a quantifier is equivalent to a formula with bounded quantifiers. In the philosophical pages I then justify this principle itself from a philosophical understanding of the difference between sets and classes: not that of limitation of size, but using a concept of mathematical time flow (through an interplay between set theory and model theory) : "a class is not a set if it remains able to contain elements which do not exist yet" (which may happen for very small classes such as one just able to contain one future element). The remarkable fact that the powerset does NOT comply to this condition, accounts for the possibility for different universes to interpret the powerset differently.
I introduce categories quite early (even before a formal definition of natural numbers). I start with a variant of the concept of concrete category, without any mention of functors. (I used the word "functor" for a more general concept which I need to name but did not see named by other authors : the structures of any theory are operators and predicates ; the unary operators are what I call functors). That is my way to give categories a good place in the picture, implicitly responding to arguments of some category theorists that "ZF is not good because it does speak about categories, so let us take categories as an alternative foundation of maths". No, categories are nice but I never saw a good way for category theory to replace set theory as a foundation of maths anyway, so here is what I see as a good balance instead.
I give a clean abstract mathematical definition of the concept of algebraic term, which I did not see well done elsewhere, as a particular case of relational system. It is not a particular case of "strings of symbols", an approach seemingly common by other authors (influenced by computer science where everything is a file that is a string of symbols ?), but which I see complicating things uselessly by its need to define criteria of syntactic correctness and then interpretations of such strings, and by its implicit use of arithmetic in the concept of "string". My way does not assume arithmetic (except for arities of symbols) but, on the contrary, provides it as a particular case of term algebra.
I give a short proof of the Completeness theorem of first-order logic : less than one page, that is much shorter than how I usually saw it done by other authors. Why do they spend so much time with complications when a simple way is possible ?
I finally explain the right philosophical justification for the axiom schema of replacement, which is much more subtle and complex than naively assumed.

For now I just have a few drafts on the foundations of geometry, which I plan to improve and develop later (affine, projective, conformal, hyperbolic... first done in French long ago). Despite the abstraction which may repel some, I found interesting to present some universal algebra (clones, polymorphisms,...) as a basis for linear algebra and duality, before developing it into the formalism of tensors. I generally optimized the expression of many topics, to take much fewer pages than usual courses for a similar amount of knowledge, by many small ideas which I cannot list here.

Foundations of physics

In the physics part of my site I provide two general overviews of physics, and a series of expositions of specific theories. I did not work on it as much as on the math yet, so it remains quite incomplete, and more of the pages are still drafts.
In the list of physical theories I explain the logical articulations between the diverse more or less fundamental theories of physics, and the hierarchical orders between them (another map of theories is given in the metaphysics text).
In the exploration of physics by dimensional analysis I present a wide overview of many phenomena by simply expressing the orders of magnitude of their characteristic quantities, such as the size of atoms and the velocities of the sound, as determined by the fundamental constants (and some contingent parameters), so as to give hints of how things work according to diverse theories without entering their actual formulation.

The exposition of specific theories starts with Special Relativity, which I see as the logically second theory of physics after geometry: it just describes the geometry of our physical universe, that is space-time, more correctly than the mere 3D Euclidean space we usually imagine it to be.
I still have to develop the exposition of relativistic mechanics from the Least Action Principle to the conservation laws (which would already need tensors for a clean and complete formulation). But I already explained the fundamental equation of General Relativity by a simple approach, already rather rigorous without the tensorial formalization (required for more general cases) based on the example of the universal expansion. This may be continued, on the one hand up to electromagnetism (I only wrote a comment on its Lagrangian), on the other hand up to the symplectic geometry of the phase space.

But I wrote what comes after this: the Liouville theorem (conservation of volume of the phase space) gives the foundation of thermodynamics. I precisely explained the nature of entropy and its creation process, by an exposition which follows the main logical structure of how it is actually known to work on the basis of quantum mechanics, but without actually formulating this basis; instead of this, I relate things with the approximation of classical mechanics. While this exposition isn't actually rigorous (since the reasoning does not match the described "classical foundation" while its actual quantum justification isn't explicitly formulated at that step), it still provides a clarity making it look somehow simpler, more intuitive and coherent than usual expositions still nowadays trying (and failing) to provide a form of logical rigor on the conceptual basis of pure classical mechanics (which was the only available one when statistical physics was first formulated, until it was found to not be the best match to reality). In particular I take account of the quantum fact that the states of systems are really probabilistic ones as they are undetermined until they are measured. And despite some widespread prejudices, this actually simplifies the math.

Then comes an introduction to quantum physics, and a review of its main interpretations. I found a way to express some basic aspects of quantum physics in mathematically accurate but quite simpler ways than usual courses, actually explaining in a simple geometric language (affine and projective transformations) the mathematical coherence of its paradoxical predictions such as those of the double slit experiment. Then I develop critical reviews of the main interpretations (Copenhagen, Bohm, many-worlds, spontaneous collapse), and an exposition of the one I support with original details: the Von Neumann-Wigner interpretation, giving a fundamental role to consciousness (considered immaterial) in the wavefunction collapse. I develop this into an original ontological position, variant of idealism : mind/mathematics dualism (admitting mind and mathematics as two distinct fundamental substances, separately time-ordered, while the physical is a combination of them). I also provide many links to references on the topic.

Other pages

In settheory.net/world I developed a long list of links to teams (and more isolated researchers) on logic and mathematical foundations around the world, encompassing 3 kinds of orientations (and thus of faculties they may belong to) : mathematics, computer science and philosophy. I guess it may be about 80% exhaustive (it becomes harder to complete as the few missing ones are harder to find). I also listed some other kinds of resources in the field (journals, proving software, organizations, blogs, conferences, courses...). It is ironical to see all researchers working in the institutions, so much focused on similar kinds of missions and the publication of their own work without caring as much about that of their peers, that none of them cared to invest as much efforts as I did into this task of listing their peers worldwide.

Another important part of my site (settheory.net/future/) is a series of texts commenting diverse aspects of the stakes of the world's future, such as the failure of communism, human intelligence, and environmental, economic, demographic, political and educational aspects. I wrote much of these in reply to many essays I reviewed from the 2014 fqxi essay contest "How Should Humanity Steer the Future?".

Finally, another big part of my research was to design a plan of new online social network which would resolve many current defects of the Web and other world's problems. In the page "Why I am upset" I commented on the dire lack of people combining the qualities of care and intelligence (ready and able to deeply and properly think outside the beaten paths and then act over the conclusions); how the few such people have no good place in this world, and the big consequence I faced: that, no matter that I could usually convince all the people who took the time to understand parts of my plan were convinced of its high value and rather cheap feasability, it still has "no reputation" just because "reputation" in this world remains a matter of global rumor, something completely mindless and circular which no mindful person ever cares to correct, as there is no good way to share and structure non-trivial information on trust and reputation. Ironically, this problem is precisely among the main ones my plan would resolve, as it was the starting issue around which I designed it. To implement it, all I need is one or a few good web programmers (a category of people usually very hard to find available at an affordable price as they are so much demanded on the world market).

Published in the book Logic around the world (cover), ISBN:978-600-6386-99-7