Just like a square becomes a cube with more dimensions, is there room for other variants of a political system that no one is really thinking about and do not fit the coordinates present at this time? — Christoffer

What about these dimensions?

1. "Level of individual participation in taking decisions" One side of the dimension: low participation, less control of the majority of people over power, but quicker process of decision making. Other side of the dimension: high participation, more control of the majority of people over power, but more difficult and lengthy process of decision making.

2. "Competition vs collaboration": One side of the dimension: high competition, better selection for giving power to the most fit ( maybe called Darwinism? not sure ) but lower ability to act as a group. Other side of the dimension: worse selection of the most fit but better ability to act as a group (altruism?)

How do these two dimensions fit in the graph? are they independent from the other two?

In physics infinite is a simplification of the measure of an object, when the real measure is big enough to be ignored. So, for example, the electromagnetic field emitted by an antenna is the superposition of an infinite number of infinitely big spherical waves. We know that this is not true (this is only an approximation), but the approximation is essential to understand the symmetries of the real field.

The same is true for geometry: Archimedes discovered that you can calculate the circumference of the circle (knowing it's ray) if you consider it as a polygon with a very big number of sides: but to describe the symmetries of the circle you cannot treat it as if it was a polygon: you have to treat it as an "infinite" polygon.

Actually, even flat geometric figures do not exist in reality, because everything in nature is 3-dimensional, and it's obvious that considering one of the dimensions to be of measure 0 is only an approximation. So, in reality, not a big deal...

The problem with infinity started to be disturbing with calculus: you need a single measure (a number) that is approximated to zero in one place of a formula an not approximated (treated as finite number) in another place: for example if you take a very big number of very little segments that form a circle, it's clear that their sum is finite (the length of the circle), but if you build a little square on top of every little segment, you get a very thin ring whose area is approximated to zero. At the end, a rational explanation of this fact was found: you have to split at the same time in all dimensions, so that at all times you get only finite objects. This is a logically coherent explanation, only that if you have a lot of dimensions to consider at the same time, you get very complex formulas because it contains a lot of terms of the "wrong" dimensionality (for example you have to sum areas with lengths), that you know you could throw away (and in reality everybody with some experience in calculus throws them away in the calculations), but to be "logically coherent" with the underlying logic explanation of limits you "pretend" to have included them in the calculations.

But the "elimination" of infinity from mathematics started to become really complicated when Cantor discovered that something similar happens with infinitely big discrete objects (instead of infinitely small continuous objects): natural numbers are not enough to enumerate all discrete sets. In fact, there is no way to enumerate the set that is made of all possible tuples of natural numbers (pairs, triples, quadruples, etc..). So, the infinite of all tuples of numbers is more infinite then the infinite of all numbers, and in the same way you can build infinitely many infinites each one bigger than the previous one.

Now, it's clear that you can't neither count the uncountable discrete sets nor measure the infinitesimally small measures, but nevertheless, these imaginary infinte models have very interesting symmetries, that happen to correspond with a surprising accuracy to the symmetries that are present everywhere in nature. Without infinites, there are no symmetries and everything is a big mess (it's something like trying to discover the laws of Euclidean geometry using only polygons).

So, nobody really knows if infinites exist in nature or what are they really like, but they are essential in mathematics to discover very elegant models that often correspond to laws of nature, and I think that the idea that mathematics should work only using finite numbers because infinites don't really exist has been abandoned forever.

Even a committed constructivist would have to acknowledge these facts, and hold the standard real numbers as important at least as an abstraction.

* As a final point, I believe as far as I can tell that not every HOTT-er is a diehard constructivist. In some versions of HOTT there are axioms equivalent to Cauchy completeness and even the axiom of choice.

If I'm understanding correctly, one need not commit to constructivism in order to enjoy the benefits of HOTT and computer-assisted proof. — fishfry

HOTT is not a constructivist theory (with my definition of constructivism) because it uses a non computable axiom: the univalence axiom (https://ncatlab.org/nlab/show/univalence+axiom).

This was considered by Voevodsky as the main "problem" of the theory, and there are currently several attempts to buid a constructive version of HOTT. One of them is cubical type theory (https://ncatlab.org/nlab/show/cubical+type+theory), but I don't know anything about it.

[ THIS WAS THE LAST PART :-) ]

* Cauchy completeness is a second order property. It's equivalent to the least upper bound property, which says that every nonempty set of reals that's bounded above has a least upper bound. It's second-order since it quantifies over subsets of reals and not just over the reals.

The second order theory of the reals is categorical. That means that every model of the reals that includes the least upper bound axiom is isomorphic to the standard reals. Up to isomorphism there is only one model of the standard reals; and it is the only model that is Cauchy complete. — fishfry

Yes, and this clarifies a lot o things about infinity:

"In first-order logic, only theories with a finite model can be categorical." (form https://en.wikipedia.org/wiki/Categorical_theory). ZFC is a first-order theory and it has no finite model (obviously), than it cannot be categorical. Ergo, you cannot use ZFC to decide the cardinality of real numbers: "if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities." (from the same page of wikipedia).

Then, you can say, the problem is in the language: let's use a second or higher order language, and you can discover the "real" cardinality of real numbers.
Well, in my opinion this is only a way of "hiding" the problem: it is true that if you assume the induction principle as part of the rules of logic, you get a limit on the cardinality of possible models (the induction principle quantifies over all propositions, so it's not expressible in first-order logic), but this is exactly the same thing as adding an axiom (in second order logic) and not assuming the induction principle as a rule. Ultimately, the problem is that the induction principle is not provable by using a finite (recursively computable) model: it's not "physically" provable.
That's exactly the same situation as for the parallels postulate in euclidean geometry: you cannot prove it with a physical geometric construction (finite model), because it speaks about something that happens at the infinite, and the fact of being true or not depends on the physical model that you use: if computers that have an illimited amount of memory do not exist, or, equivalently, if infinite topological structures do not exist, then the induction principle is false, and infinitesimals are real!
So, the sentence "if you use uncomputable (non constructive) axioms in logic, you can decide the cardinality of real numbers", for me it sounds like "if you use euclidean geometry, you can prove the parallel postulate".

The standard reals are the only model of the reals that are Cauchy complete.
The constructive reals are not complete — fishfry

I tried to google for "constructive real numbers are not complete", or something similar.
I found this, for example: https://users.dimi.uniud.it/~pietro.digianantonio/papers/copy_pdf/RealsAxioms.pdf ).

And I found this: https://www.encyclopediaofmath.org/index.php/Constructive_analysis

I think this is what you refer to by "constructive reals". Is it?
Can you give me a link where is written that they are not complete?

I am convinced that my definition of "constructivism" is not the same thing that your definition.

Well, here's a simple definition of what I mean by constructive logic:
=== A logic is called constructive if every time that you write "exists t" it means that you can compute the value of t. ===

I believe that you can define real numbers that are complete in a constructive logic. I think the example that I gave you using Coq is one of these. But I could be wrong: I am not completely sure about this.

On the other hand the other famous alternative model of the reals, the hyperreals of nonstandard analysis, are also not complete. Any non-Archimedean field (one that contains infinitesimals) is necessarily incomplete. — fishfry

I googled this: "non archimedean fields are not complete" and the first link that come out is this one:
https://math.stackexchange.com/questions/17687/example-of-a-complete-non-archimedean-ordered-field

Probably, as they say, "The devil is in the detail". I read several times in the past about Abraham Robinson's hyperreal numbers, and I believe that I read somewhere that non archimedean fields are not complete. So I believe that, under appropriate assumptions, this is true. But why is this a problem?

The constructive reals fail to be complete because there are too few of them. The hyperreals fail to be complete because there are too many of them. The standard reals are the Goldilocks model of the real numbers. Not too small and not too large. Just the right size to be complete. — fishfry

Hmmm... I understand what you mean:
- "constructive" reals are computable functions. Then there is a countable number of them.
- standard reals are the set of all convergent successions of rationals then their cardinality is aleph-1
- nonstandard reals are much more than this (not sure about cardinality), since for each standard real there is an entire real line of non-standard ones.

Well, here's how I see it:
- "constructive" reals (with my definition) can be put in one-to-one correspondence with standard reals, only with a different representation (but I don't know a proof of this) and do not correspond to computable functions. It is true that if you can write "Exists x such that ... " then you can compute that x, But for the most part of real numbers x there is no corresponding formula to describe them (and this is exactly the same thing that happens for non constructive reals).
- Robinson's nonstandard reals are more than the standard reals because you exclude induction principle as an axiom (so that "P(0)" and "P(n) -> P(n+1)" does not imply "forall n, P(n)"). But there are objects used in mathematics that are treated as if they were real numbers, but DO NOT have the right cardinality to be standard real numbers: for example the random variables used in statistics: https://en.wikipedia.org/wiki/Random_variable . So, they are more similar to nonstandard reals.
- The real numbers of smooth infinitesimal analysis are less then standard real numbers, and even the set of functions from reals to reals is countable: basically, every function from reals to reals is continuous and expandable as a Fourier series. And there are infinitesimals.
What for such a strange thing? Well, for example, they correspond exactly to what is needed for the wave-functions and linear operators of quantum mechanics: there are as many functions as real numbers, and a real numbers correspond to experiments (then, there are a numerable quantity of "real" numbers). And what's more important, a wave function contains a definite quantity of information, that is preserved by the laws of quantum mechanics.

So, from my point of view, there is not one "good" model of real numbers, at the same way as there is not one "good" model of geometric space.

[ END OF PART TWO :-) ]

Arghhh. In my opinion you are totally missing the point of math. I don't mean that to be such a strong statement, but in this instance ... yes.

The point isn't to have a pristine logical proof. The point is the beautiful lifting, via the axiom of choice, from the very commonplace paradoxical composition of the free group on two letters, up to a paradoxical composition of three-space itself. It's the idea that's important, not the formal proof. That is the actual point of view of working mathematicians.

I recently ran across an article, link later if I can find it. Professional number theorists were asked by professinal logicians, wouldn't you be interested in seeing a computerized formal version of Wiles's proof of Fermat's last theorem? And the logicians were stunned to discover that the mathematicians had no interest in such a thing! — fishfry

First of all, the point about formal systems. I completely agree that the formal proof is not the essential part of a theorem about geometry! I'll tell you more: I am pretty convinced that the formal proof alone does not contain the essential information necessary to "understand" the theorem, and I started to write on this forum because I was looking for somebody that has some results/ideas on this point. Please take a look at my last post on the discussion that I opened about two months ago: "Is it possible to define a measure how 'interesting' is a theorem?" (https://thephilosophyforum.com/discussion/5789/is-it-possible-to-define-a-measure-how-interesting-is-a-theorem).
Then, since nobody seemed to find the subject interesting, I started to reply to posts about infinity.. :-)

But the point of the discussion about infinity was more or less this one: "are ZFC axioms about infinity right?" and my answer was: "you cannot use a formal logic system to decide which kind of infinity exists". And I said you that I "don't like" ZFC because, for example, of Banach-Tarski paradox.

Let me try to explain this point: I am not saying that ZFC is wrong and Type Theory, or Coq, is right.
I am saying that the encoding of segments (and even surfaces and solids) as sets of points is not "natural", because you can build functions that have as input a set of definite measure and as output a set where measure is not definable: the transformations on sets and on measures are not part of the same category! Of course, you can say that this is because there really exist geometric objects that are not measurable, and it is really a feature of geometry, and not of ZFC.
Well, in that case you have to admit that there exist several different geometries, because there are sound logic systems based on type theory where all sets are measurable and lines are made of a countable set of "elementary" lines (integrals are always defined as series) (elementary lines have the property that their squared length is zero), and Banach-Tarski is false.
So, we can ask which one of the possible geometries is the "right" one. The problem is that they are only mathematical models. From my point of view, this is exactly the same thing as mathematical models for physics (or maybe you want to consider euclidean geometry as a model, but in that case non measurable sets surely do not exist): the best mathematical model is the one that encodes the greatest number of physical results using the smallest number of physical laws, and none of them is perfectly corresponding to physical reality anyway.

What's important about Wiles's proof is the ideas; not every last bit and byte of formal correctness.

This is something a lot of people don't get about math. It's the overarching ideas that people are researching. Sure, you can work out a formal proof if you like, but that's more like grunt work. The researchers are not interested.

Likewise, you are interested in a formal computer proof of BT, and that is not the point at all. The point is that the paradoxical decomposition of the free group on two letters induces a paradoxical decomposition of three space. That's the point. It's beautiful and strange. Formal proof in Coq? Ok, whatever floats your boat. But that is not the meaning of the theorem. The meaning is in the idea. — fishfry

I was looking for a formal proof of BT in ZF Set theory because the critical part of BT is the function that builds the non measurable sets (the one defined using of the axiom of choice). You are right that in most cases the formal logic is not necessary at all, but in this case the result of the theorem depends in a critical way on the actual encoding of a continuous space as a set of points. In fact, I am pretty convinced that It's possible to encode euclidean geometry in ZF Set theory in a way that is perfectly sensible for all results of euclidean geometry but makes BT false (I think I can do it in Coq, but surely you wouldn't like it.. :-) )

If you don't like the idea of "space as a set of points" I do understand that philosophical objection. But then you are objecting to a huge amount of modern math and physical science too. — fishfry

That is not true: the definition of measure is purely algebraic, and can be used with ANY definition of sets: it's the same thing as for calculus, derivatives, integrals, etc..

I'm not sure where you're coming from. If you don't like point sets, then you wouldn't like set theory. I can certainly see that. — fishfry

Well, I think you can easily guess: I studied electronic engineering (Coq can be used to proof the correctness of digital circuits), but I am working as a software developer. But I am even interested in mathematics and physics, and even philosophy, so I think I am not the "standard" kind of software developer.. :-)

I didn't say that I don't like point sets: I said that point sets are not a good model for 3-dimensional geometric objects, and I am not very original with this idea: in HOTT segments are not sets but 1-spaces. The word "sets" is defined as synonymous of 0-space (or "discrete" space).

But surely your objection then is not to BT, but to virtually all of modern mathematics and physics. Is that your viewpoint? How far does your rejection of using point sets to model mathematical ideas go? — fishfry

Point-set topology and algebraic topology are equivalent, and all modern mathematics and physics since Grothendieck (https://en.wikipedia.org/wiki/Alexander_Grothendieck) are based on category theory: as you just said, they really don't care much about foundations... :-)
Both string theory and loop quantum gravity (the two most trendy at the moment....) (https://en.wikipedia.org/wiki/Loop_quantum_gravity) make heavy use of algebraic topology: not only they don't care "of what are made" the objects of the theory, but they are even not clear if they are something geometric, or maybe can be interpreted as emergent structures coming from something even more fundamental. This is the point of view of category theory: don't describe how objects are made, but only how they relate with each-other.

By the way you don't need the power of the well-ordering theorem to change the order type of a set via bijection. Just take the natural numbers 0, 1, 2, 3, ... and reorder them to 1, 2, 3, ..., 0. So it's the usual order but n '<' 0 for every nonzero n, where '<' is the new "funny" order relation. The natural numbers in their usual order have no greatest element; but in the funny order they do. We've changed the order type with a simple bijection. You don't argue that this simple example should be banished from decent mathematics, do you? The funny order is the ordinal number ω+1ω+1, or omega plus one. Turing knew all about ordinals, he wrote his Ph.D. thesis on ordinal models of computation. Even constructivists believe in ordinals. — fishfry

I was speaking about changing the topology of a set: any open set (that has no minimum or maximum) can be transformed into a set with a minimum (lower bound), not open and not closed. If the isometries used in BT are continuous transformations, they should preserve the topology of the sets that are transformed. My guess is that the ones used in BT cannot be continuous on sets that are not discrete (that is one of the things that I wanted to understand from the formal proof..). In other words, I think that to make that transformation you have to destroy the topological structure of the object (but I am actually not sure about this). If this wasn't possible (and, as I understand, it's not possible without axiom of choice) I think BT would be false.

It's can't be right or wrong any more than the game of chess can be right or wrong. BT is undeniably a theorem of ZFC. That's true even if you utterly reject ZFC. The novel Moby Dick is fiction, yet it is "true" within the novel that Ahab is the captain of the Pequod and not the cabin boy. Banach-Tarski is a valid derivation in ZFC regardless of whether you like ZFC as your math foundation. — fishfry

Absolutely, I agree.

Suppose I stipulate that ZFC is banned as the foundation of math.

Ok. Then Banach-Tarski is still a valid theorem of ZFC. So what is your objection to that? B-T is still fascinating and strange and its proof is surprisingly simple. One could enjoy it on its own terms without "believing" in it, whatever that means. I don't think the game of chess is "true," only that it's interesting and fun. Likewise ZFC. — fishfry

Yes, it's even more interesting because of the fact that it is an (apparent) paradox, and paradoxes are the most informative and interesting parts of mathematics.

AHA! Yes you are right. But these are RIGID MOTIONS. That is the point. We are not applying topological or continuous transformations in general, which can of course distort an object. — fishfry

Well, I have my doubts here. These are rigid motions on countable subsets of points, it means only on subsets with zero measure. I am quite sure that they cannot be continuous transformations on measurable subsets with non-zero measure. If you have a proof that they are, I am very interested in it ( even not formal :-) )

P.S. Algebraic definition of measure: see https://en.wikipedia.org/wiki/Sigma-algebra

The Wiki article on the subject has a nice outline in case anyone's interested in demystifying this theorem for themselves. Point being that BT is much less esoteric than people think. — fishfry

OK, I read the article and finally understood the point about isometry group! :smile:

Actually, I tried to look for a formal proof of Banach-Tarski in ZFC Set theory, but the only one I found is using Coq... :-) (https://hal.archives-ouvertes.fr/hal-01673378/document)

- By the way, there is a formal logic computer system using ZFC Set theory with a very extensive library (http://mizar.org/), but there is no Banach-Tarski theorem there (at least at the moment)

I am looking for formalized proofs because usually proofs in ZFC related real numbers are too long to be completely written by hand, and tend to be not clear on the steps that make concrete use of ZFC axioms.

Anyway, the reason of my doubts on ZFC in geometry are more "fundamental", and not related to the existence of the isometry group.
The thing that I don't like is that in BT (as in all topology based on ZFC), a segment (or a surface, or a solid) is DEFINED to be a set of points.
(well since in ZFC everything is defined as a set, you don't really have much choice with this definition...)

The problem is that you can't use the property of a segment being a set in any way, because equivalence relations on sets (one-to-one functions) do not preserve any of the essential properties of the segment:
- the size of the set is not preserved, because additivity doesn't work for uncountable sets (and I believe there is no way to define a sum of an uncountable set of terms that gives a finite result, because the elements of an uncountable set are not "identifiable" with a computable function, and then you can't "use" them to compute a sum).
- the topology of the set is not preserved, since every set can be reordered to have a least elenent (https://en.wikipedia.org/wiki/Well-ordering_theorem).

For this reason, the use of bijective functions with the "meaning" of moving a 3-dimensional object by breaking it into pieces and then reassembling it in another place is wrong.
I mean: it's not contradictory, but it doesn't reflect the intuitive meaning that you give to the transformation, since neither topology nor measure are invariant under this transformation.

The "trick" of breaking the transformation in two parts, so that the second part is only the composition of four pieces doesn't change the fact that the first part of the transformation does not preserve measure (the measure of each of the four intermediate pieces is undefined).

[TO BE CONTINUED]

It is exactly this kind of visual puzzles that Immanuel Kant rejected in his Critique of Pure Reason. Classical Euclidean geometry is not pure reason, because it requires solving visual puzzles. Immanuel Kant was adamant: Pure reason is language only. It may only make use of symbol manipulation. — alcontali

Agreed. Pythogoras theorem is only surprising in a visual puzzling environment such as classical Greek geometry. — alcontali

Yes, and that's the point: you can't see the reason why Pythagoras theorem is interesting if you don't look at physical space. Then, we can say Pythagoras theorem is an interesting fact about physical space, and you can't "understand" it looking only at the logical (language only) formulation of the theorem, or even looking at it's proof. I guess a contemporary geometer would say that the exceptional fact is that the Riemann curvature tensor of physical space is zero (at least with a very good approximation).

You can of course restrict mathematical proofs to only make use of symbol manipulation. And that is the same thing as restricting proofs to use formal logic (more or less the standard of modern mathematics): a proof is a finite set of applications of a finite set of rules acting on a countable set of symbols.
But in my opinion even the use of "pure reason" relies ultimately on the ability to perform experiments in the physic world, only of a more "fundamental" level than the ones performed in euclidean geometry.
In fact, ultimately, a proof verification presupposes the existence of computable functions, and that presupposes the ability to perform a physical "experiment" that operates on a discrete set of distinguishable inputs and outputs, and is supposed to return always the same output when is given the same input. According to quantum mechanics, it's even not possible to build such an experiment with absolute certainty, since laws physics are not deterministic. So, I wouldn't attribute to "pure reason" a fundamentally non-physical character, as opposed to the physical character of Euclidean geometry.
In fact, Homotopy Type Theory has proved the equivalence between Homotopy Theory and Type Theory, and the existence of a model for both theories based on simplicial sets. So, HOTT formal logic corresponds to the use of topology on simplicial sets, that probably can be seen as the most fundamental laws of geometry.

Experimental testing is forbidden in math. Therefore, this approach by number sampling is very, very un-mathematical. In fact, you do not prove anything by sampling lots of numbers. The proof must be the result of judicious symbol manipulation instead. — alcontali

I am not saying that proving is based on number sampling. I am saying that understanding the meaning, or the importance of a theorem, is based on number sampling or, more generally, on performing experiments on (imaginary, and very often geometric) models. And that's exactly what is missing in a formal system such as Coq. What I am saying is that the expression of a theorem and it's proof in a formal logic system simply DOES NOT CONTAIN THE INFORMATION necessary to understand it, at least for a great part of the traditional "important" theorems (not all of them). And what mathematicians use in reality is information that is not contained in the formal (purely syntactical) expression of the theory, but is expressed in words and explanations written in plain english. So, a function (or an artificial intelligence system) capable to compute the "importance" of a theorem cannot be based on the logical formulation of the theory alone, but needs to take into account the model (often a physical model) on which the theory is interpreted.

There is an obvious reason why ancient Greek mathematicians regarded the irrationality of square root of two as a paradox: irrational numbers do not exist in nature! For them a length is a physical entity and a measure is a process made of splitting the segment in smaller and smaller parts until you find it's measure (expressed as a fraction). Infinite splitting is not a physical process. Then, irrational numbers cannot possibly exist as physical entity. This deduction is impossible to be made on the sole base of the formulation of the theorem "square root of two is irrational" (even by an human mathematician), simply because the information about the interpretation of fractions as physical measure is missing.

For polynomials with coefficients in the rationals, you can indeed still reduce an otherwise irreducible polynomial by adding the appropriate field extension. Still, even adding i does not guarantee a closed-form solution for the roots (constructed using only supported field operators: + - * /), because we are not sure that there is a tower of radical field extensions available to achieve that. Even though adding i guarantees a solution, it may not spare you from having to approximate. — alcontali

Yes, of course. I was considering a polynomial with coefficients in the real numbers. In this case the field extension obtained adding "i" corresponds to the field of complex numbers. I usually omit lots of details in my arguments, to make them shorter.

Indeed, no. Gödel's incompleteness precludes that. It is generally not possible to discover new theorems by enumerating the domain of theorems and then verifying if they are provable in the theory. — alcontali

I agree. I don't think this is due to Gödel's incompleteness. But I wasn't speaking about enumerating the domain of theorems. Actually, I wasn't even speaking about finding proofs automatically, but only defining a kind of "measure" of the relevance of a theorem based on the formulation of the theorem in a formal system AND it's interpretation on a model (that, in current mathematical practice, is usually not expressed in a formal language). The ability to recognize the relevance of theorems is obviously fundamental for a search strategy for theorems, but is in my opinion something completely different from enumerating proofs. It's somehow similar (but not the same thing) to searching the tree of positions in a chess game as opposed to evaluating a position: a function to evaluate positions is essential to a game strategy, but is not based on a blind simulation of the game. Understanding a theorem is not based on proof searching.

In my impression, the surprising element is that you do not need to extend it further to guarantee a solution for all polynomials with rational coefficients, but not necessarily in closed form. — alcontali

Yes, I agree. In fact, the theorem doesn't assert the existence of closed forms for solutions (obviously not true), but is based on the intuition that polynomial functions are continuous and surjective.

I think that the practice of using field extensions more or less guaranteed that i would be discovered ... — alcontali

Yes, but I would say that the algebraic concept of field extension is understandable once you have some concrete examples of field extension (as imaginary numbers). Algebraic concepts are usually generalizations of properties that several concrete modes have in common.

Let me give you some simple examples of theorems taken from here: http://www.cs.ru.nl/~freek/100/ to illustrate my idea of "probability measure" for a theorem.


1. Pythagorean Theorem

- Theorem expressed in a formal language: (Theorem number 4 from https://madiot.fr/coq100/)

Theorem Pythagore :
forall A B C : PO,
orthogonal (vec A B) (vec A C) <->
Rsqr (distance B C) = Rsqr (distance A B) + Rsqr (distance A C) :>R.


From a formal point of view, it's a logical equivalence between orthogonality of two vectors and an equation between real numbers (vectors' lengths).

Should an equivalence of this kind be surprising? (or improbable)? The answer is NO: in fact, whatever angle you take between two vectors, there will be an induced equation between vectors' lengths, and vice-versa.

So, why do you find the theorem surprising when you read it for the first time if you didn't know it before? (if you understand it)
Because it's a coincidence between two special cases:
- orthogonality is a special case of angle between vectors (a very particular one: the length of the projection of one vector on the other is zero)
- the sum of squares is a special case of equation: it's very simple, and made only of primitive operations addition and squaring.

So, could a computer program recognize this theorem as surprising (or improbable)? Yes, but only if you give to the program a model where the coincidence between the two special cases occur.
A model could be of course a computer-vision software plus the ability to paint triangles and observe them, but could even be some much simpler linear algebra software.
The essential feature here is to recognize zero as a special case of real number and addition and squaring as special cases of mathematical functions.
And that is easily achievable if you have an axiomatization of real numbers and you measure the complexity of a number or function simply by using the length of their definition (taken as Shannon information measure)

In other words, both terms of the logical equivalence have a very small complexity (or information measure) if interpreted taking as model an axiomatization of real numbers.


==========================================================================

2. The Irrationality of the Square Root of 2

- Theorem expressed in a formal language: (Theorem number 1 from https://madiot.fr/coq100/)

Theorem sqrt2_not_rational : forall p q, q <> 0 -> p * p <> q * q * 2.


From a formal point of view, it says that a particular equation (p * p = q * q * 2) has no solution on the domain of natural numbers.

Should an equivalence of this kind be surprising? (or improbable)? The answer is NO: infact, most of diophantine equations have no solutions.

So, why do we find the theorem surprising?
Because square root of two is the measure of the square of side one, and every measure in the physical world is represented by a fraction (because measuring is a finite physical process: you can never find square root of two as a result of a measure in physics).

Then, if we give a computer a "physical" model of geometry (for example a world made of pixels), it could recognize, experimenting with it, that every length can be measured, and every measure is in fact a fraction. So, it could recognize as "surprising" (or statistically very improbable) a theorem that says that there is a measure that is not a fraction.

==========================================================================

3. Fundamental Theorem of Algebra

- Theorem expressed in a formal language: (Theorem number 2 from http://www.cse.unsw.edu.au/~kleing/top100/ - COQ version is very long if I include the necessary definitions)

lemma fundamental_theorem_of_algebra:
assumes nc: "¬ constant (poly p)"
shows "∃z::complex. poly p z = 0"


From a formal point of view, it says that every equation of a certain type (non constant polynomial) has always a solution on a certain domain (complex numbers).
Why should this be an exceptional case in comparison with a lot of other types of equations that always have solutions on certain domains?

First of all, a polynomial is a function that has a very simple definition (in the sense of low complexity): a polynomial is whatever function you can build using only "times" and "plus" operations with one variable. This fact alone makes it an interesting object.
A simple concept (as for complexity of it's definition) related to every function is it's inverse. So you easily discover that not all polynomials are invertible.
The surprising thing (that makes the theory more interesting), as we know, is the discovery of complex numbers: if you only take one polynomial that has no solution (the simplest one: "X * X = -1") and "pretend" that it had a solution by adding a new symbol "i" to the number system, then you discover that now all polynomials have a solution.
Complex numbers then become even more interesting when you discover that they correspond exactly to a SIMPLE geometric model (other coincidence).

Would a computer be able to "discover" complex numbers? I don't think so (at least not using the technologies currently used for artificial intelligence). But I think that a computer could be able to RECOGNIZE that complex numbers are an interesting concept by using a measure of how "improbable" are theorems when interpreted on the right models (meaning: the models that people have in mind when they say that they "understand" the theorem).

Evidently he thinks that Iran will not respond to an attack starting a war. Otherwise it doesn't make much difference how many people you kill with the first bombs.

But he said that his decision was based on the number of people that will die in the attack. Does anybody believe this?

I just read this, reported to be a tweet from Trump:
""...On Monday they shot down an unmanned drone flying in International Waters. We were cocked & loaded to retaliate last night on 3 different sights when I asked, how many will die. 150 people, sir, was the answer from a General. 10 minutes before the strike I stopped it, not proportionate to shooting down an unmanned drone..""

Well, the question is obvious: how many people are the equivalent of an unmanned drone?

OK, probably you are right: my question was too generic and didn't express what I wanted to mean.

Actually, I changed idea about the "No" answer since i wrote that post :razz: (2 months ago)

I think that the case of mathematics is quite different from judgments on subjects such as, for example, music or literature.

In fact, I believe that between mathematicians that work on a given theory there is quite unanimous consensus on which theorems are the most important ones.

There are of course lots of people thinking that the whole subject of mathematics is not interesting at all. But between the group of people that understand a given theory, the definition of "interesting" can be very often (if not always) translated to "surprising", or "improbable coincidence". And that's the kind of judgment that, in my opinion, can be "measured" in an objective way.

Of course there will be people that disagree with this "measure", but we have a quite objective test for how good this measure would be: the history of mathematics is full of theorems that have been discovered and then forgotten, because judged irrelevant, and a very few of them that have been studied in schools for hundreds or even thousands of years.

I don’t know but i think work on measuring abstract human concepts like how interesting, original, artistic, aesthetic and so forth things are should be useful in the work on developing artificial intelligence — Frotunes

Yes, I agree.

But I also suspect it too will suffer from what psychology and ethics suffer from, which is that are rather vague, and what humanities suffers from, which is that they’re rather generalising and socio-political, or what formal logic suffers from, which is that it’s rather dismissive of plain common sense (which, though sometimes prone to irrationality, is often very useful) — Frotunes

Well, I believe mathematics is a different case in this regard: I think that the degree of "interest" of a theory can be defined, somehow similarly to how the degree of "randomness" of a number has been defined.

Physical objects have properties in common: numbers are a property of objects that are made of separate parts, and then can be counted. Names are possible results of an algorithm that creates all possible strings made from a given alphabet. If something can be defined in a precise way, it means that there exists some kind of "attribute" common to different physical objects that identifies the abstract object. These "attributes" are identifiable information that is contained in physical objects, and information "exists" in reality.

OK, now I see what you mean. You are saying that maybe there is no meaning in saying that a particular string of characters, or a particular number "exists", if nobody has never written or thought about it in some way. Well, I am convinced Platonist. I do think that all possible numbers, or strings, "exist" in some concrete sense, even if nobody ever thought about them, or even when human beings didn't exist yet on earth.

Well, first of all, I am happy that finally somebody considered my question interesting :smile:

I actually have a quite different opinion about this: I think that provability, or how easy is it to find a proof, has nothing (or very little) to do with the fact that the theorem expresses an interesting mathematical proposition. But I agree with you that some kind predictability measure (such as the one defined by Shannon information theory) should have a role.
My opinion is that the essential information is not in the theorem itself, but in it's "interpretation" on the mathematical model that we have in mind. In fact, I believe that even if you took a very talented mathematician that has no knowledge at all about a new theory, and you presented him with the purely formal expression of a theorem, he wouldn't be able to say if that is something interesting or absolutely boring. In fact there are several cases in the history of mathematics of original results that were completely ignored at a first time, and then become very important after several years from their publication: one famous example is Galois' group theory.
So, from this point of view, the representation of a theory in Category theory as a functor from the logical proposition to the the model should be the object that is taken into account to calculate Shannon's probability. And in my opinion there exists an objective way to define such a function in a way that matches very closely what mathematicians consider to be the degree of "interest" of a theory. And that, of course, would be essential to build some kind of A.I. software that "understands" mathematics.

Well, for proper names that denote a particular object, I think somebody must have assigned a name to the object before you can use it, so the answer is no. For variables of logical propositions, they are only arbitrary strings or arbitrary length (at least in formal logic), so there is no need that somebody assigned a meaning to the name before using it.

Maybe I didn't understand what you mean by a "name". I was thinking about names used in logic propositions, that are simply meaningless labels.

In this case surely there are proper names that no one has said or thought: just take a random string of 30 letters (and maybe add some vowels to make easier to pronounce): there is a very high probability that nobody has ever thought or said that name before!

No, names used in logic are simply strings of characters: formal logic is a purely syntactic system.

You could imagine to use geometric objects (whose sizes are supposed to be an uncountable set) instead of strings for names, and use geometric constructions as rules. In that case you would have a "logic" based on an uncountably infinite set of "names", but then the problem of recognizing if two names (or lengths) are the same I think would become undecidable: there is no physical way to compare an uncountable set of lengths to decide if they are the same.

Suppose I take the set of infinite lists of ones and zeros. I know from Cantor's diagonal that this is uncountable. So I give each its own name; are there then uncountably many proper names?

First-order predicate logic apparently assumes only a countable number of proper names: a,b,c...

How would it change if there were an uncountable number of proper names? — Banno

The point is that logic derivations have to be of finite length. So you can never use more than a finite set of names in a formal proof. Even if you imagine to have an uncountable set of names, the set of names that you can use in any derivation, however complex, will always be countable. So, it doesn't make any difference what's the cardinality of set of names that you have. The only thing that counts is the cardinality of the set of names that you can use in a derivation.

Here is a link: https://www.ourdocuments.gov/doc.php?flash=false&doc=90&page=transcript — Fooloso4

Really very interesting. Thanks for the link!

I think they want to eliminate any threat to the United States from anywhere in the world. — Fooloso4

I think what their policy has exactly the opposite effect: the threat to the United States is increased from Russia and China as a result of keeping their nuclear arms constantly in state of alarm. Exiting from ICBM missiles treaty the time of reaction when a missile launch is detected is reduced to minutes now, and there are increasingly more nations that have ICBM missiles pointed to the US and ready to launch. So the risk of a nuclear war is surely much bigger now.

Surely US military is perfectly able to estimate the risk caused by politicians' choices, so I don't believe they care too much about the increased threat to the United States

Perhaps only if Iran does not do what they want. Like Bush's "Mission Accomplished" they may believe that all that is needed is a show of power. — Fooloso4

But what do they want exactly? That the religious leaders leave their places to people chosen by US? How should Iran regime be changed? What should they do to avoid war?

OK, but they can't be paid directly from arms producers, right? I mean: this would be illegal ( or not? ).

They are vested in the profit yielded from selling the means to kill people. — ernestm

Do you mean they earn from selling arms? What is "the profit yielded from selling the means to kill people" ?

OK, I am more inclined to believe this. But what is their personal advantage concretely?

But do you think they really believe that to achieve peace and resources for Iranian people you have to make war that will destroy the country?

No, I am saying that this seems to be Bolton's position. You asked why do Bolton and Pompeo want a war with Iran. — Fooloso4

OK, I was speaking about Bolton's position too. So, let me reformulate it:

Bolton's position is that the self-interest of the United States is to dominate the region by military force, so that there will be peace, stability, freedom, and democracy, and not to give up anything in negotiations, to be able to impose their will without any concessions.

What goes on in the Middle East is a matter of our self-interest. Instability in the region has global economic impact. — Fooloso4

That's the thing that in my opinion doesn't make sense: you are saying that US trying to dominate the region by military force to ensure them freedom and democracy.
How can Iranians be free and have democracy if they will be dominated by a foreign by military force? Isn't this an obvious contradiction?
Let's suppose that, after loosing a war against US, Iran will become a democratic state. Well, the first thing that they would vote for (if they really were a democracy and were able do decide for themselves) would be to get rid of the domination of US!
You can't allow them to have freedom and democracy, if you want to dominate the region. Isn't it obvious?

OK, so you say that the self-interest of the United States is to dominate the region by military force, so that there will be peace, stability, freedom, and democracy.
This sounds as an altruistic motivation: the United States have to spend their money and their soldiers to ensure peace, stability and freedom for people on the other side of the world.
I would say that the self-interest of US (or at least the self-interest of the citizens of US) is exactly the opposite: they should care only about their own peace, stability, freedom, and democracy.
What you are describing sounds more like an altruistic interest, and not self-interest.

I have a simple question: why do Bolton and Pompeo want a war with Iran ? (because it seems that they are trying to provoke a war with Iran, right? Or are they only trying to protect the world from the growing danger that comes from Iran?).

I mean: assuming that they want to go to war with Iran, what's the reason? Does anybody has a convincing explanation, different from the silly one that "that they are bad people"? A lot of people says that there is an economic convenience for them if America goes to war with Iran. If this is true, how do they earn money from this exactly? Only by selling arms? They are not the owners of arms industries, right? So how do they get the money exactly? And if it's not about money, then what's their motivation? Personal ambition? Making the world a better place? or what?

I heard a lot of different opinions about these issues, but nobody never gives a convincing explanation of how the whole system works, and what are people's motivations to act as they do.

Well, I see that you are not speaking of mathematics or logic here. So I think I have not much to say in reply to your answer...

But in my opinion the "infinity" in mathematics is not such an arcane entity as your "one" thing.
Infinitely big and infinitely small are the "simplification" of extremely big and extremely small.
So for example you can consider an infinitely long and infinitely thin line as a model of a very long and thin wire, because you want to "abstract away" the properties related to it's real length and width, and consider only the position and orientation of the wire.
The only reason that infinite numbers were ruled out from mathematics (on the contrary of infinite geometric objects, that were always considered allowed) was the idea that you cannot reason about them without getting inconsistencies in logic. But now we know how to deal with infinity using formal logic in a perfectly sound way, and nearly all important mathematics is making use of infinite and infinitesimals.

So, to me asking if infinite really exists is like asking if a plan really exists, since all "real" objects are 3-dimensional: it's only a mathematical approximation (simplification) due to the fact that you want to ignore one of the 3 dimensions. But probably this point of view is only due to my limited knowledge about philosophy..

if you don't allow the existence of infinite sets, you have to treat segments as a different kind of thing than a set of points.
But in the galois field, they treat the segment made of points but don't use infinite sets, the one mentioned in the article.Can you send me any article,book recommendations that make your position clear to me cause l may be confusing your point here, I hope not. — Wittgenstein

Sorry, I realize now that I didn't answer to this question.

I read the article that you posted (https://plato.stanford.edu/entries/geometry-finitism/supplement.html).
It's not true that "they treat the segment made of points". They say that "a point p corresponds to a couple (x,y)" and "A line corresponds to a triple (a,b,c)"; they don't say that "a line is a set of three points". But I agree that this is an example of finite models for Euclides' axioms, (by the way, in their original form Euclides' axioms are not expressed in a formal language, but I don't want to be picky on this point).

The infinite model that I was referring to is the standard one, based on the standard topology of the real line (https://en.wikipedia.org/wiki/Real_line).
I actually don't have a proof that there are no models of Euclides' axioms where a line is a finite set of points. On the contrary, I think you can easily build one by taking as space a finite 2-dimensional array of points, but this is obviously not the right model for the physical space.
So, probably I should have said that: if you want to build a model of the physical space where a segment is described as a set of points (as in standard topology), you need infinite sets. Otherwise, you have to build a model of the physical space where a segment is not a set of points (an example of this is kind, not based on set theory, is smooth infinitesimal analysis: https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis).

OK, so I can prove that 1 = 2. Here's the proof:

I take a segment of length one and I stretch it until it becomes of length 2. But this is still the same segment, so 1 = 2. What's the difference between this and your argument? The halves of two are the same as one. Why is your argument not valid for finite lengths?

I can even prove that a segment the same as a circle: just bend it and it becomes a circle!

If I want to compare the size of two objects I can operate on the objects only with transformations that don't change their size: for example I can put them one on top of the other by moving them, but I can't split them in two or stretch them.

I think I didn't understand completely your argument, but why can't you say the same thing for two instead of one?

Yes, but we are speaking about measuring the length of segments. You can't stretch the segment if you want to measure it.

Which of the two questions do you answer? both?

So, you say that infinite lines in euclidean geometry are the same thing as segments of length one, right?

I thought this is obviously absurd. OK, if I have to say why: because infinite lines always intersect if they are not parallel, and finite segments can be not parallel and not intersect

What about infinite lines in euclidean geometry? Are they allowed? Are they the same thing as segments of length one?

I tried to look for the "standard" encoding of category theory in HoTT, but it seems that there are many, and none is standard (same thing as in set theory...). Anyway, the one chosen in the HoTT book is not what I said: objects are not types, but there is a type whose elements are the objects of the category.

Anyway, the idea of representing groups with functors is very simple:
first of all, since isomorphisms are equivalence relations, UA implies that isomorphic groups are equal. So, as you said, in the category of groups there is only one object representing the cyclic group of order 4.
But of course you can build objects that have the structure of a group (group objects) in any cartesian closed category: for example you can define natural and complex numbers in the usual way on the category of sets (surely in HoTT there are many much more efficient representations of natural and real numbers than this one but, as long as you can prove that they are isomorphic, the one that you choose is only a matter or convenience).
So a representation of the category of groups with sets is a functor from Groups to Sets. And of course there are many different representations of the same group as sets, corresponding to many different functors from groups to sets.

(https://ncatlab.org/nlab/show/representation):
"3. General definition
In a rather general form, we therefore have a representation of a category C in a category D is simply a functor F:C→D."