No, mathematics has quantitative relations as its subject matter — Dfpolis

Number theory is no longer the dominant axiomatization in mathematics, and has not been for over a century (Dedekind-Peano). Nowadays, it is set theory that is considered the dominant axiomatization in mathematics (ZFC, aka, Zermelo-Fränkel-Choice).

In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.

Number theory is not even Turing-Complete, and hence, considered to be a relatively weak and incomplete axiomatization.

Besides set theory, there are numerous other Turing-complete axiomatizations such as function theory (lambda calculus), type theory, combinator theory, and so on. Every Turing-complete axiomatization is capable of expressing all possible knowledge in its associated language.

Numbers are just one type of building brick in mathematics. Sets, functions, types, combinators, and so on, are other types. You can trivially express numbers as sets, with e.g. Von Neumann ordinals:

0 = { }, 1 = {{ }}, 2 = {{ }, {{ }}}, 3 = {{ }, {{ }}, {{ }, {{ }}}}, and so on. Therefore, numbers are not a separate building brick in set theory -- that would be unnecessary -- but are just set expressions.

Hence, mathematics is not just about quantities, or numbers, which are not even essential in math. Mathematics is the set of all theorems that you can derive using the axiomatic method from any consistent axiomatic system. Number theory is merely one such axiomatization.

The answer is simple: a scientifically valid ToE makes concrete predictions about what will happen. — Theologian

Muslims believe that the divine destiny is when God wrote down in the Preserved Tablet ("al-lawh al-mahfooz") (several other spellings are used for this in English) all that has happened and will happen, which will come to pass as written.

Quran, Sura 57, ayat 22. No calamity befalls on the earth or in yourselves but is inscribed in the Book of Decrees (al-lawh al-mahfooz), before We bring it into existence. Verily, that is easy for Allah.

The Preserved Tablet is a device that contains the decrees, i.e. the finitary set of axiomatic rules that perfectly predict the future. Therefore, the Preserved Tablet is the Theory of Everything (ToE). The Quran insists that God has a copy of it.

When someone says something, don't just accept it. Ask, "Is this correct? What's the argument or evidence for it — Terrapin Station

What we see, i.e. the input signals we receive, create some kind of model in our heads, i.e. an abstraction of the physical world. With all complex abstractions being leaky, this process inevitably, occasionally produces unexpected results, i.e. situations where the perception as a model is out of sync with what it is trying to model.

We are not necessarily aware of when these perception errors take place. Therefore, I agree with Kant that we have no guarantee that the "appearance" that we see is a faithful representation of the thing-in-itself. In that sense, it is legitimate to declare the thing-in-itself to be an unknown.

Hence, you can even argue in favour of Kant's unknown thing-in-itself view using the law of leaky abstractions, with perception itself being the leaky abstraction.

Prolegomena, § 32.And we indeed, rightly considering objects of sense as mere appearances, confess thereby that they are based upon a thing in itself, though we know not this thing as it is in itself, but only know its appearances, viz., the way in which our senses are affected by this unknown something.

Maybe if you folks stopped treating Kant like a religious messiah. — Terrapin Station

I just think that Kant is often surprisingly spot on. I do not believe that he was infallible or so ...

I agree that ideas are not physical, but I rather prefer a dualist interpretation, whereby humans are able to interface between the Platonic realm of abstractions, and actual objects, to produce neat things like: — Wayfarer

Say that L is the set of all possible expressions in language, then Lr is a subset of L in which the language expressions seek to be isomorphic with the real, physical world R.

I agree that:

Lr ⊂ L and Lr ≈ R

Let's call Lr "the map" and R "the territory".

One major problem is, of course, that R is actually unknown. As Immanuel Kant famously quipped: Das Ding an sich ist ein Unbekänntes. (The thing in itself is an unknown).

We often use Lr and R interchangeably, and that is often no problem, but in the cases in which it is a problem, we may soon run into an abstraction leak, because ultimately the map is not the territory:

The map–territory relation describes the relationship between an object and a representation of that object, as in the relation between a geographical territory and a map of it. Polish-American scientist and philosopher Alfred Korzybski remarked that "the map is not the territory" and that "the word is not the thing", encapsulating his view that an abstraction derived from something, or a reaction to it, is not the thing itself. Korzybski held that many people do confuse maps with territories, that is, confuse models of reality with reality itself. The relationship has also been expressed in other terms, such as Alan Watts's "The menu is not the meal."

As coined by Joel Spolsky, the Law of Leaky Abstractions states:

All non-trivial abstractions, to some degree, are leaky.

Not only do abstract models not represent reality at all, unless you painstakingly expend effort to maintain such isomorphism, they do not even need to do so, in order to be useful. Mathematical axiomatizations, for example, never represent reality, while theorems must not be considered to be mathematical unless they belong to such axiomatization. They could be something else, however; such as scientific, for example.

In other words, a theorem can be mathematical or can be scientific, but can never be both at the same time.

I suggest you read Roger Penrose, The Emperor's Mew Mind, in which he shows that human minds are able to solve uncomputable problems. — Dfpolis

I assume that the book is copyrighted, but I have still found a summary.

Gödel's procedure is actually just a special case, i.e. merely an example, of the fact that human minds are able to solve uncomputable problems. Gödel proves that there is no knowable procedure possible to discover new knowledge. Still, humanity obviously did it. So, yes, just the discovery of new knowledge is already one such uncomputable problem.

In other words, it will indeed never be possible to explain (as in knowledge) why humanity has managed to discover its existing stock of knowledge. If the human brain were just some kind of biological computer, it would not have been possible at all.

Furthermore, there is something in that human brain that allows us to decide the otherwise (computationally) undecidable problem whether God exists or not. At the same time, there is absolutely no input that you could ever feed to a computer, short of the undiscoverable ToE (Theory of Everything) that will allow it to decide this question.

As metamathematical proofs do not belong to mathematics, so metaphysical proofs do not belong to physics. — Dfpolis

The terminology is confusing in this regard, because metamathematics is defined as a subdivision of mathematics, while metaphysics is defined as non-physics.

Metamathematics uses the same axiomatic method as mathematics. The reason for the meta is because the objects studied are mathematical theories themselves. It concerns axiomatic theories about other axiomatic theories.

Metaphysics does NOT use the same scientific method (of experimental testing) as physics. Hence, physics is a subdiscipline of science, but metaphysics is not.

So progress (or lack of progress) toward a physical ToE is entirely irrelevant. — Dfpolis

The ideal of the ToE is to discard the scientific method, i.e. experimental testing, and be able to do science using the axiomatic method, i.e. proving by axiomatic derivation. The reason why science is not axiomatic, is because the axiomatic base for physics is lacking.

Hence, science would like to be like mathematics, but does not have the instruments available to do so. Science and its method of experimental testing is therefore some kind of poor man's mathematics. Science does not use the scientific method because it wants to, but simply because the desired alternative, i.e. axiomatic provability, is not attainable.

I think that Godel's work has little to say about a ToE, because the method of physics is not the method of mathematics. Physics is not built on a closed axiomatic foundation, but an open experiential foundation. That does not make a ToE possible, but it does make the analogy with mathematics highly questionable. — Dfpolis

The ToE is exactly about replacing the scientific method by the axiomatic one. Stephen Hawking explores this possibility at length in his lecture, Gödel and the End of Physics.

About the ToE, i.e. the ultimate (mathematical) theory of the universe in terms of a finite number of principles (=axioms), Hawking said: What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted.

Some people will be very disappointed if there is not an ultimate theory that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind.


Hawking believed that the ToE is not attainable because of Gödel's incompleteness theorems.

Again, you are confusing methodologies. Natural science uses the hypothetico-deductive method, while metaphysical proofs often use strict deduction. — Dfpolis

Well, the ToE is exactly about replacing the one by the other, and the very reasons why this is not possible. In the discourse on the ToE, the confusion is simply deliberate.

On the one hand, you claim we can prove nothing about reality, and, on the other hand, you seem to claim to have proven that we can prove nothing about reality -- which is proving something about reality. — Dfpolis

This impossibility does not prove anything about the real world, but about the relationship between us and the real world. It just means that we do not have access to a copy of the axioms from which the real world has been/is being constructed.

This shows a fundamental misunderstanding of the nature of knowledge as awareness of present intelligibility. Knowledge is new if previously unactualized intelligibility is actualized by awareness. Yes, what is new may have been implicit in existing knowledge, but that only means that it was intelligible, not that it was actually known — Dfpolis

The nature of knowledge as a justified (true) belief, JtB, requires that it has the shape of an arrow. If Q can be justified from P, then Q is knowledge. Having access to Q is insufficient. It is not knowledge, until the necessity of the arrow, i.e. the justification, has been demonstrated.

You consider Q to be knowledge in and of itself. That is contrary to the Platonic definition, i.e. JtB, which places knowledge in the arrow between P and Q. If knowledge cannot be written in arrow format, P=>Q, then it is not knowledge. In that case, Q is merely a conjecture.

This is a complete non sequitur. Just because what we already know can be the basis of some new knowledge, does not mean that it can be the basis for all possible knowledge. New knowledge can come both from reflection on what we already know and from new types of experience, e.g. the kinds of observations and experiments that have informed science since antiquity. — Dfpolis

Well, this is exactly what Gödel tries to achieve in his incompleteness theorems. He maps encoded knowledge statements onto numbers, and their justification, i.e. proof, onto other numbers. Then, he investigates if he could enumerate all numbers that constitute formally valid language and verify if it has a corresponding number representing proof.

The simplest way to figure out if it could work, is to try to solve Turing's halting problem by enumerating all possible programs, and verify if they actually halt, i.e. if their computation stops:

Assume that we have a consistent and complete axiomatization of all true first-order logic statements about natural numbers. Then we can build an algorithm that enumerates all these statements. This means that there is an algorithm N(n) that, given a natural number n, computes a true first-order logic statement about natural numbers such that, for all the true statements, there is at least one n such that N(n) yields that statement.

We already know that such procedure cannot exist. Therefore, if Gödel's procedure exists, then it would solve Turing's Halting problem too. That is exactly what is not possible. Hence Gödel's procedure cannot possibly exist. In other words, even though you can represent all knowledge theorems as numbers, you cannot just enumerate all possible numbers to discover all possible knowledge.

Let's consider the second incompleteness theorem, which rules out self-proofs of consistency. If we have a set of axioms that are not merely posited, but properly abstracted from reality, we do not need to prove that they are consistent, because no contradictions can be instantiated in reality. That means that simultaneously instantiated axioms have to be mutually consistent. It is only if one restricts the knowledge base to an abstract system that there is a problem. Opening ourselves to reality can often resolve such problems. — Dfpolis

That is a very constructivist remark (constructivism).

In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists.

Traditionally, some mathematicians have been suspicious, if not antagonistic, towards mathematical constructivism, largely because of limitations they believed it to pose for constructive analysis. These views were forcefully expressed by David Hilbert in 1928, when he wrote in Grundlagen der Mathematik [...]

Even though most mathematicians do not accept the constructivist's thesis that only mathematics done based on constructive methods is sound, constructive methods are increasingly of interest on non-ideological grounds.

Constructivism is a long debate. To cut a long story short, I consider constructivism to be heretical.

That's just one view. Another view is that there is no separate "abstract, platonic world," yet we still have logic, here in the real world — Terrapin Station

There is a view that the linguistic expression "chair" most certainly is physical, as is everything else. That view is called "physicalism." — Terrapin Station

Ok, understood.

I can only say that it is not the dominant view in mathematics, which is staunchly Platonic. There is a fringe philosophy, called constructivism, in mathematics that goes in that direction, but it is generally considered to be a heresy.

Traditionally, some mathematicians have been suspicious, if not antagonistic, towards mathematical constructivism, largely because of limitations they believed it to pose for constructive analysis. These views were forcefully expressed by David Hilbert in 1928, when he wrote in Grundlagen der Mathematik [...]

Even though most mathematicians do not accept the constructivist's thesis that only mathematics done based on constructive methods is sound, constructive methods are increasingly of interest on non-ideological grounds.

I tend to follow David Hilbert's view on (real-world) constructivism, for similar reasons, and I am therefore also very negative about it. I think that the constructivist mentality is unproductive. Therefore, I consider it to be a heresy.

How is a language expression not a real-world phenomenon? The essence of ostensive definition is language binding itself to the real world. — Pantagruel

I can only disagree.

What about Game of Thrones? Doesn't the television series bathe in language? Does it even pretend to be about the real world? It is obviously completely imaginary!

Therefore, language has generally nothing to do with describing the real world.

Some subset of the set of language expressions does indeed attempt to be isomorphic with the real, physical world, but that is just a subset. You can do much, much more with language than merely describing the real world!

If I utter the phrase "Pick up that stone" and you pick it up, how is that not a real-world phenomenon? — Pantagruel

Yes, in that particular case, the language expressions are about real-world, physical objects, situations, events, and other observables.

I have never said that language cannot attempt to describe the real world. It sometimes even moderately successfully does so, even though it always remains a simplifying abstraction, albeit one that is somehow isomorphic with the real, physical world.

I have only said that not all language expressions attempt to do that. Language does not need to refer to the real world. For example, a language expression can trivially refer to other language expressions.

You don't have to buy anything a la platonism to do proofs. — Terrapin Station

In fact, the term "Platonic" is just a figure of speech to refer to an abstraction, i.e. a mere language expression. I just use it to distinguish them from physical, real-world objects. So, a chair is a physical object, but the language expression "chair" is not.

There is a simple litmus test for platonicity of the target of a language expression.

If you can translate it into other languages, then it must be a language object. For example, "5" is a language object, because you can also write "five", "cinque", "fünf", or "101" (binary). Therefore, it has nothing to do with the real, physical world. It is an idea instead of something physical.

Till this very day, Platonist philosophy is the core of the philosophy of mathematics. It is not literally Plato, though. It is just highly Plato-like, and it is clearly inspired by Plato's work.

My own take is somehow related to Edward Zalta's Platonic abstract object theory.

In my opinion, however, Zalta fails to mention explicitly the fact that mathematical (abstract-Platonic) objects do not exist outside the context of the axiomatic system ("axiomatization"), i.e. the construction logic, in which they are defined.

Proof theory requires the prover to supply the axiomatization in which the proof is derived.

Therefore, I do not say that this would be an error in Zalta's theory, but I consider this certainly to be an omission (or a lack of emphasis) in his otherwise excellent characterization of abstract objects as encoded language.

So part of the background assumptions you're working with is that the physical world has different (and unknown) logic? You'd need to support that claim. — Terrapin Station

Logic does not operate on real-world observables. It operates on statements, which are not real-world, but language objects that live in their own abstract, Platonic world.

Logic are not statements about the real-world. Logic consists of statements about other statements.

Since when is a language expression a real-world phenomenon? Does it have energy, mass, gravity, or an electrostatic charge? What could there possibly be physical about a language expression? Hence, a language expression cannot possibly be an object in the real, physical world.

How are you arriving at that conclusion? — Terrapin Station

Proof is context-sensitive.

It is only valid in the abstract, Platonic world in which it necessarily follows from its construction logic.

For example, you can prove a theorem from number theory, or from set theory, or from the lambda calculus (and so on). These are three different axiomatizations, i.e. abstract Platonic worlds, with each their own set of theorems.

None of their theorems prove anything about the real, physical world. The real, physical world has another (unknown) construction logic.

Furthermore, none of the theorems of one abstract, Platonic world (=axiomatization) proves anything about another abstract, Platonic world.

The real problem is that the proof is in violation of proof theory.

Proving a theorem amounts to demonstrating that it necessarily follows from the explicitly-stated axiomatic construction logic of the abstract, Platonic world in which it is provable.

God is defined as the creator of the real, physical world.

Therefore, to prove the theorem, we would need access to the axiomatic construction logic of the real, physical world, also called, the theory of everything (ToE).

In his lecture, Gödel and the End of Physics, Stephen Hawking quite successfully argued, however, that we cannot possibly discover the ToE. Gödel's incompleteness theorems prevent us from achieving that feat.

This implies that it is not possible to prove anything at all about the real world. It is not possible to prove that anything exists, and science does not prove anything about the real world.

Since you cannot prove anything about the real, physical world, you cannot prove anything about its creation.

This does not mean that God exists or does not exists. It only means that our knowledge methods fail to reach the answer to this question. Hence, the belief that God exists or does not exist is necessarily the result of something else than knowledge.

There is nothing special about that, actually.

For example, access to existing knowledge is insufficient for the purpose of discovering new knowledge. Therefore, the most important ingredient in the discovery process of new knowledge is something else than knowledge. Otherwise, our existing knowledge would allow us to enumerate all possible knowledge theorems, and use that to discover new knowledge. That is exactly, however, what Gödel's incompleteness theorems disallow.

You can also prove this impossibility from Turing's halting problem. In fact, that is how you can trivially produce an alternative proof for Gödel's incompleteness theorems from Turing's halting problem (for the weaker form).

When the human subject masters reality by means of reason and/or science, the self comes to understand itself as existing in a fundamentally nominative mode. That is to say, the self becomes the subject that applies the disciplines of reason and science to the world, which is thereby conceived to be the object of that activity. — Matias

It was probably quite predictable that a bout of scientific and technological progress would make some people, and even cultures, much more arrogant than they used to be.

Just like Moore's law on exponential growth in computer CPU speed was inevitably going to come to an end some day -- it already did at least a decade ago -- economic growth and growth of per capita income will also soon disappear. The flurry of impressive new discoveries in nuclear physics, for example, also came to an end at least fifty years ago, and nobody is sending people to the moon any longer.

We could easily be entering an era of mere stagnation. Still, the future could even be worse than that.

Quire a few short-term decisions in the style "after me the deluge" are increasingly getting burdened by their long-term consequences.

For example, most of the existing nuclear plants around the globe are about to reach the end of their useful lives. At that point, they will no longer make money, but they will still need to be decommissioned at enormous expense. I can guarantee that it is not the people who have made lots of money from these nuclear plants when they were still operating, who will pick up any of the decommissioning cost. That problem will have been trivially predictable. You could see it coming from the get-go, but that will not make any difference as to what is inevitably going to happen, and who exactly will be on the hook for the gigantic decommissioning bills.

The worst arrogance can undoubtedly be found in the social sphere. People no longer need children, because the government will take care of them in their old age. And where is the government going to find the resources to do that? Well, obviously, from other people's children!

So, there is a hell of a lot of fundamentally arrogant stuff that is simply not going to keep flying. We can probably not afford positivism.

So, could a computer program recognize this theorem as surprising (or improbable)? — Mephist

You have (at least) two possible starting points. You can start from Euclid's classical axiomatic basis in geometry, or else axiomatize from number theory (Dedekind-Peano) by using a coordinate system. Euclid is much harder to reason from because it amounts to visual puzzling.

So, we can start from three two-tuples, say in a simple Cartesian coordinate system -- you can pick any coordinate system, really, and consider {(x1,y1),(x2,y2),(x3,y3)}. We can simplify by moving the triangle to the axes as following {(0,0),(0,yA),(xA,0)}. The angle in (0,0) is guaranteed to be perpendicular by the coordinate system itself. You can then trivially verify from the definition of distance=(x2-x1)²+(y2-y1)² that the distance between the two points that are not the origin. is xA²+yA². With every right-angled triangle isomorphic with the shape produced by picking a a point on each axis along with the origin, Pythagoras theorem easily follows from mere number theory. Instead of 2 points, you can pick 3,4, or more points, along with the origin and then derive other, similar distance invariants.

A model could be of course a computer-vision software plus the ability to paint triangles and observe them — Mephist

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.

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. — Mephist

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

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

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.

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. — Mephist

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.

Would a computer be able to "discover" complex numbers? I don't think so — Mephist

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.

But I think that a compute 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 — Mephist

Well, you can solve a increasing number of polynomials with rational coefficients just by extending the rationals with judiciously chosen radicals. Therefore, it was probably not a far stretch to attempt to extend the rational calculation field with the positive solution of x²=-1 and give that symbol a name. 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.

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

If Socrates said it then yes — Frotunes

I suspect that Socrates himself would have rejected the view: "Everything Socrates ever said, must be considered philosophical, because it was Socrates who said it."

If Socrates ever said, "I seem to have an indigestion today", then reporting to have an indigestion has now also become part of philosophy.

Therefore, it can obviously only be exactly the other way around. Some of the statements Socrates made were philosophical and therefore, we consider Socrates to be a philosopher.

There are two kinds of people. People who believe propositions because of whom said it, and people who believe propositions because of how they were said (i.e. their justification).

The "believe-who" people are known as the populace, who are trivially manipulated and misled, and who should not be allowed to make any serious decision that affects others. They cannot verify a justification, and therefore have to trust. That is obviously an accident waiting to happen.

But I agree with you that some kind predictability measure (such as the one defined by Shannon information theory) should have a role. — Mephist

Yes, I agree. Furthermore, we do not really need to solve the problem.

We would only need to show some progress, and give them impression that the approach is possibly promising.

That could actually be enough for writing a grant proposal and get funding to do some work on the problem. There are institutions that actually pay for this kind of stuff. If you manage to remotely justify why they should hand over resources from their precious budgets, they may even do it.

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. — Mephist

If you look at the set of axioms A of a theory T and then at a theorem S that is provable from T, the surprise is probably related to some concept of distance d(S,A). The longer the distance, the more surprising, I guess.

It also depends on the existence of other known provable theorems S1,S2,..., and the distance d(S,S1), d(S,S2), and so on. If d(S,S[k]) is large for each known-provable S[k], then theorem S could be deemed "interesting".

So, maybe we could try to develop a notion of theorem "distance"?

It has to be some kind of "shortest distance", because otherwise, people could make their theorems more interesting just by adding otherwise useless steps in their proof.

o, 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. — Mephist

Yes, we could consider necessary non-superfluous steps in the proof to be a concept for distance, and hence somehow some kind of proxy for Shannon probability?

And that, of course, would be essential to build some kind of A.I. software that "understands" mathematics. — Mephist

Maybe some kind of plugin or script for Coq or Isabelle?

So how could that be applied in such an abstract 'possibility space', so to speak? — Wayfarer

Yeah, that is why a highly simplified strategy is easier to implement. Check if the theorem is already known to be provable, and then it is not interesting. It is only interesting to the system, if the system cannot determine if it is provable, but you manage to supply the proof. All existing known provable theorems are then considered boring by the system.

My question is: is there a sensible way to define a measure how 'interesting' is a given theorem (or theory)? — Mephist

In my impression, we would need a function F that returns the a priori likelihood that any arbitrary theorem s is provable from theory T. From there on, we can use Shannon information theory to compute the surprisal, i.e. the information content, i.e. the self-information of the message that says that s is indeed provable from T. We could simply use proof P as the message here.

So, a simple strategy consists in constructing the best possible software program that tries to look up the existing proof for any arbitrary, given theorem. If the program can locate this proof, then there is no surprise attached to the fact that the theorem is provable, and then the theorem is not particularly interesting. If the program cannot locate the proof, but you still manage to supply it, then your theorem actually is interesting.

owever, the statement, " an infinite number of elements " is a contradiction in terms as infinity is clearly not a number — Wittgenstein

In the viewpoint that math is about symbol manipulation formalisms, we may not even be interested in the concept "meaning" as some kind of correspondence factor with the real, physical world.

For a starters, the symbol ∞ is just one character and not an ever-growing sequence of characters. Hence, it is perfectly suitable for participation in symbol-manipulation formalisms. An ever-growing sequence of characters, however, would be a problem, because in that case our symbol-manipulation algorithm may not even terminate.

Secondly, the symbol ∞ is not irrelevant or immaterial, because it is definitely mentioned in usable reduction rules that can successfully extend classical arithmetic. For example: a + ∞ = ∞ and also a * ∞ = ∞, with for example, a ∈ ℝ. The symbol clearly has some kind of "absorbing" effect on other elements in its domain. You can actually obtain/generate the ∞ symbol by performing particular manipulations on more common elements of the domain, such as 1/0 = ∞.

With the various reduction rules available, the symbol ∞ could actually be useful when you seek to produce a closed form output result from a particular input expression. The domain does not even need to be numerical and the algebraic structure not necessarily a field. The rule templates will undoubtedly still be consistent.

For example, with a and b arrays, elements from the Array domain, and a + b defined in a particular meaningful way, and ∞ a meaningful extension to the Array domain, you will find that a + ∞ = ∞.

So, it will still absorb the other elements during addition and multiplication. In that sense, it is a bit similar to the zero symbol, which also absorbs other elements after multiplication while leaving them unchanged after addition.

In other words, a field or other algebraic structure can successfully be extended with the symbol ∞ while maintaining consistency and while satisfying the pattern in existing rule templates for the symbol. So, infinity may indeed not be a number but it is certainly a legitimate extension element in numerical algebraic structures.

OK, but he clearly misunderstood Gettier because Gettier didn't conclude that knowledge is justified belief. Gettier only concluded, in his own words, that the JTB definition "does not state a sufficient condition for someone's knowing a given proposition." — Michael

His argument is surprisingly closer to the examples Gettier gave, than it looks like. Gettier manually concocts what physicists call an entanglement, and then concludes, "Oh My God, it goes wrong now!".

Well, these Gettier entanglements naturally occur all the time in quantum physics. No need to fabricate them!

That is why he is not really interested in Gettier's actual examples, which even look relatively silly compared to the ones that spontaneously occur in nature.

Who wrote this? Clearly not Gettier. Gettier himself didn't conclude that knowledge is justified belief. — Michael

This person is clearly someone with a background in physics.

Werner Heisenberg already pointed out in 1927 a massive flaw in the correspondence-theory truth about the position (p) and velocity (v) tuple of an electron: (p,v).

An electron may indeed have a (p,v) tuple, but if you know one element in the tuple, you can impossibly know the other element. That is his notorious uncertainty principle. Heisenberg received the Nobel prize in 1932 for arguing this. It forced everybody in nuclear physics to switch over to quantum mechanics, to the utter dismay of Albert Einstein, who hated this.

Schrödinger's cat is another gigantic problem with the correspondence theory of truth. The state of a particle is indeterminate until the external world observes it:

According to Schrödinger, the Copenhagen interpretation implies that the cat remains both alive and dead until the state has been observed.

The correspondence theory of truth simply starts falling apart at the level of nuclear physics. In fact, both Heisenberg and Schrödinger rediscovered what Kant had already insisted on: Das Ding an sich ist ein Unbekänntes.

In other words, what Gettier argued in his own work sounds trivially obvious to nuclear physicists.

I did not use the physics argument, because I do not think like a physicist. I came to the same conclusion by looking at the nature of mathematics. That is a completely different route, but one that suits me much better than physics, which I certainly recognize as a sound discipline, but one that I do not really like, because I prefer the abstract, Platonic worlds of mathematicians to the real, physical world of scientists.

That's not Gettier's conclusion. — Michael

I did not make that conclusion directly based on Gettier's work or examples, but on the argument in the debate.org discussion about JTB:

He wrote:

Simply put" Justified belief" is enough. As stated so clearly by Gettier, it is possible for a proposition to be simultaneously true and false in a similar way as Shrodingers cat can be both alive and dead before obtaining incontrovertible evidence to prove it is one or the other. Knowledge does not equal truth so to add that into the definition of knowledge is tenable.

As I wrote, I came via another route to the idea that "Justified belief" is enough. A mathematical theorem is a justified belief that is not correspondence-theory true. So, with mathematics clearly being considered knowledge, this requirement of correspondence-theory truth is unsustainable.

Furthermore, Immanuel Kant's synthetic statements a priori are also knowledge without any reference to sensory input or the real, physical world. Hence, synthetic statements a priori are impossibly correspondence-theory true.

Math consists of symbols made meaningful solely by virtue of our attribution. — creativesoul

The meaning of symbols comes from their relationship with other symbols. It is not possible to attribute anything to them because they often do not correspond to anything in the real, physical world. For example, what meaning could we possibly attribute to the S and K combinators in the SKI calculus? They are mentioned in the reduction rules. That is all there is to them.

All symbolic notation is existentially dependent upon common parlance. — creativesoul

You can write the symbols in full too, as complete words. They often mean absolutely nothing in common parlance. What would be the common parlance for a "tower of radical field extensions" in the Galois correspondence? There is absolutely nothing at all that corresponds to this in the real, physical world.

Math is utterly irrelevant to the role that truth plays in all human thought/belief. — creativesoul

Agreed. It has no direct incidence on correspondence-theory truth.

The reason why math plays an outsized and even dominant role in science and even in daily life, is because math is consistent by design, while the real, physical world is deemed to be consistent by assumption. The requirement of consistency forbids particular things from happening in the real world. All mathematical models in applied math or science exploit this consistency correspondence.

Math is irrelevant to this discussion — creativesoul

Math is knowledge, but math is not correspondence-theory true. Hence, math is a justified belief but not a justified true belief. Therefore, it raises exactly the same problem as the one Gettier raised.

Hume works from the unspoken premiss that reason is somehow existentially independent of emotions. — creativesoul

I'd rather agree with Hume. Reason is just a tool suitable for propositional inference. Even a machine can verify whether a new proposition is indeed provable from a set of other propositions.

Discovering new knowledge propositions is something else. We certainly do not exclusively use reason or knowledge for that. If we had knowledge on how to discover new knowledge, we would actually already have the new knowledge, and then would not need to discover it.

All reason is existentially dependent upon emotion. — creativesoul

It may originate from there somehow, but once you program the propositional inference engine in software, it no longer has anything to do with emotion. The Isabelle reasoning engine is not emotional.

In particular, Isabelle's classical reasoner can perform long chains of reasoning steps to prove formulas.

Thus... there is no such thing as 'Pure Reason' except and aside from being the name of a product(figment) of the Humean imagination. — creativesoul

Pure Reason even exists in software.

Gödel was critiquing inductive logic/reasoning. — creativesoul

There used to be this presupposition that if a proposition is (logically) true, there must necessarily exist a proof for it, somewhere to be discovered. It is, in fact, the essence of David Hilbert's Entscheidungsproblem.

Back then, at the end of the 19th century and until the 1930ies, people were obnoxiously positivist and scienticist. Some people still are today, actually. It is also the era of the arrogant God of gaps ideology and the ugly modernist Corbusier buildings, because hey, soon, we will know it all.

So, in the 1930ies, Gödel, Turing, and Church set out to prove that none of that would ever happen. It led to the notions of Gödel incompleteness and Turing-complete knowledge, which is the maximum knowledge that can ever be attained.

I reject Kant's notions of a posterior and a priori. — creativesoul

Well, Kant just defines these things: a priori meaning without the use of sensory information and a posteriori, the opposite of that (i.e. empirical).

All thought/belief are existentially dependent upon sensory experience. — creativesoul

Mathematics is pure symbol manipulation, i.e. language expressions. It does not take any sensory input. Therefore, it is pure reason. Kant criticized the practice in classical geometry (Euclid's Elements) to solve visual puzzles. So, he considered it not to be pure reason. Nowadays math is pretty much algebra only. So, Kant's issue with math has been addressed.

I knew I didn't have Godel quite right, but the gist(I thought) was that he was critiquing inductive logic/reasoning. — creativesoul

Gödel was criticizing Bertrand Russell's Principia Mathematica and its impossible optimism. It gave the wrong impression that mathematics would some day be able to solve all problems. Quod non.

Is that not a problem then? Math is all about coherence. — creativesoul

Yes, but not coherence in the real, physical world. It is about constructing abstract, Platonic worlds that are coherent by design.

Sounds like you're a mathematician — creativesoul

Not in a professional sense. Just as a hobby. I would never join the academia. They require a growing amount of ideological orthodoxy from their staff, which is unattainable for me.

physicist or something? — creativesoul

In many ways a physicist (=scientist) is the exact opposite of a mathematician.

A physicist seeks to create models that as much as possible satisfy the correspondence theory of truth. A mathematician never does that, because doing that is a constructive heresy in mathematics.

Mathematics never says anything about the real, physical world. It only says things about the abstract, Platonic world specifically constructed for that purpose; which is obviously never the real, physical world.

Therefore, mathematics and science are opposite and strictly exclude each other.

I have made my money in software engineering, but now I am semi-retired. I've recently started spending seven hours per day on physical exercise. I might come out of retirement some day, but possibly also not.

I reject Kant's notions of a posterior and a priori. — creativesoul

These are just definitions: a priori is without making use of sensory information, while a posteriori is just the opposite, i.e. empirical.

All thought/belief are existentially dependent upon sensory experience. — creativesoul

Symbol-manipulation formalisms are devoid of sensory experience. They are just blind operations on language symbols. It is what Immanuel Kant considers to be pure reason (=no sensory input). In fact, Kant originally deemed mathematics not to be pure reason, because (Greek) geometry (Euclid's Elements) revolves around solving visual puzzles. In the meanwhile, mathematics has completely abandoned the use of visual puzzles. Everything revolves around symbol manipulation, i.e. language only procedures. So, nowadays, unlike in Greek antiquity, it is pure reason.

I knew I didn't have Godel quite right, but the gist(I thought) was that he was critiquing inductive logic/reasoning. — creativesoul

Gödel was critiquing Bertrand Russell's impossible optimism in his Principia Mathematica, which gave the absurd impression that mathematics would eventually be able to provably answer all possible questions. It worked like a red cloth on Gödel, who sought to demonstrate the fundamental limitations of Russell's formalisms.

Don't use the quotation marks. — creativesoul

Yeah, I hadn't noticed the quote button when you select a fragment.

Is that not a problem then? Math is all about coherence. — creativesoul

Yes, but never in the real, physical world. Math constructs an abstract world of which the implementation guarantees coherence. It is not hard to get coherence, when you simply force the matter. It would not be possible to do that in the real world.

Godel show the that all inductive/axiomatic logic is incomplete, as you've indicated. — creativesoul

That is not exactly what Gödel proved. There are (simple) axiomatic systems that are complete, such as the Presburger arithmetic.

Gödel did the following. He constructed a virtual/abstract machine, with associated language, that is capable expressing the rules of the (more complex) Dedekind-Peano arithmetic (this is our ordinary arithmetic, actually).

He created a numerical bytecode for this virtual/abstract machine to map these language expressions into numbers. Hence, properties about these language expressions became properties of numbers. At that point, Gödel used that language to express a statement that is (logically) true but not provable, i.e. a so-called Gödel statement or Gödel number:

S = "S is not provable" ---> the bytecode is powerful enough to express this

S is indeed not provable, because there is no way in which you can derive that from the Dedekind-Peano axioms. Therefore, what S actually says is (logically) true.

Hence, there cannot possibly exist a procedure to enumerate all possible valid and logically true statements (i.e. numbers that correspond to a logically true valid language expression) and decide if the theorem that the statement represents, is provable in that system, yes or no.

It also shows that (logically) true is not the same as provable, and even that the fact that a statement is (logically) true does not imply that it will be provable in any way.

Therefore, any axiomatic system that embodies the axioms of number theory or anything else that requires a language of similar complexity, is not possibly complete, aka: It is incomplete.

Math is not knowledge. Math is method. — creativesoul

Well, in Immanuel Kant's lingo, as in Critique of Puree Reason, mathematical theorems are synthetic (=knowledge-increasing) statements a priori (=do not make use of sensory experience) that are derived axiomatically from a construction logic of analytic statements a priori.

Hence, a math theorem is an arrow:

P => Q

with P the axiomatic construction logic
with Q the theorem that necessarily follows from these axioms

Therefore, Q is a belief justified by P.

In other words, a mathematical theorem satisfies the restricted Platonic definition for knowledge, i.e. a justified belief.

Logic is not a measure of truth. — creativesoul

Agreed. Logic has nothing to do with correspondence-theory truth. Logic is just a bit of lattice algebra. It is a symbol-manipulation formalism that has nothing to do with the real, physical world. It lives in its own abstract, Platonic world, just like all mathematics.

Logical proofs prove coherency. Coherency is not enough for truth. — creativesoul

Agreed. I also complete subscribe to Bertrand Russell's criticism on the coherence theory of truth.

Gettier said nothing about these concerns. They're irrelevant. — creativesoul

I agree that Gettier said nothing about these concerns. He attacked the "true", the "T" in JTB in his particular way. I have attacked it in another way. I have just pointed out that the "T" in JTB is not sustainable in mathematics either. Not one mathematical theorem satisfies the correspondence-theory requirements for the term true. Mathematical theorems are simply not isomorphic with the real, physical world, if only, because they live in their own abstract, Platonic world.

That's not what he's saying. The typical approach to the Gettier problems, by the way, is not that JTB is wrong, but that it needs to be better qualified. — Terrapin Station

Well, with the entire field of mathematics not being correspondence-theory "true", the "T" in JTB is simply too much of a problem. If math is knowledge, then JTB is wrong. It must be JB instead.

There's a certain amount of irony here. I mean do you live somewhere else, aside from the real physical world? — creativesoul

Our conversation takes place in a virtual world. It is just an elaborate simulation of artifacts of the real, physical world. The "page" you see in front of you is not physical. It is virtual. It is not fully an abstract, Platonic world, because it still requires running processes on multiple computer systems. It is still close, though. You could see things on that screen that are totally imaginary but look completely real. These things will not be real. They will still be virtual.

The real, physical world is just one of the many worlds we operate in. The Platonic and virtual worlds in which we operate are equally relevant; sometimes even more relevant that the real, physical world.

I happily operate in all three types of worlds.

I can know that I'm typing on a computer. — creativesoul

What you see with your own eyes is true, if you are eyes are not lying to you, but it is never provable.
Try to (objectively) prove that you are typing on a computer. You cannot. It is simply impossible to do that.

This link is a good explanation as to why it is not possible to prove anything about the real, physical world.

Correspondence is easily provable. That's what verification/falsification methods look for. — creativesoul

Provable, according to proof theory, means that the proposition necessarily follows from the construction logic of the world (in which you prove it).

You cannot possibly achieve that with the real, physical world.

It just cannot be done.

That is why science, for example, which merely experimentally tests in the real, physical world, does not prove anything. Contrary to popular belief, there is no such thing as a scientific proof. General philosophy also does not prove anything.

It is not possible to prove anything at all about the real world.

At best, you can collect some kind of evidence.

That is why insisting on the correspondence theory's truth for knowledge claims is such a faulty proposition. Provable knowledge is never (correspondence theory's) true.

I'm granting all this. It's true by definition. Different animal altogether. — creativesoul

Worse than that!
It is not even true!

It is merely provable (from the construction logic of that abstract, Platonic world).

Furthermore, provable and true are entirely distinct concepts.

The correspondence theory's truth is absolutely never provable, simply, because we have no access to the (axiomatic) construction logic of the real, physical world, i.e. the notorious theory of everything.

Furthermore, Gödel's incompleteness theorems proves that there are statements that are (logically) true but not provable.

Of course, logically "true" does not mean correspondence theory's "true".

Logically "true" just means that there is an abstract, Platonic world in which statements are being mapped on a set of arbitrary truth values, one of which is arbitrarily called "true"; while these truth values satisfy a given set of rules, as specified in lattice algebra.

In other words, Gödel's incompleteness does not mean anything in the real, physical world, but it is definitely provable in the abstract, Platonic world of number theory.

I disagree. We cannot know a falsehood. — creativesoul

Statements that have no correspondence with the real, physical world are not (necessarily) false. They are simply not "true" as meant in the correspondence theory of truth.

For example, if we construct an abstract, Platonic world in which there are two symbols, S and K, and two rules [1] Kxy = x [2] Sxyz = xz(yz), then we can trivially demonstrate by applying both rules that (SKx)y = y. Therefore, SKx is the identity operator in this abstract world.

Is the claim about the identity operator "true"? (In terms of the correspondence theory)

No, because the SKI combinator calculus corresponds to absolutely nothing in the real, physical world. It is just a system of rules that create a new abstract, Platonic world.

The proposition that SKx is the identity operator is certainly provable in the SKI Platonic world, as it can be justified through standard symbol manipulation. Hence, it is a justified belief (JB), i.e. knowledge. However, it has absolutely nothing to do with the real, physical world or its associated "truth".

Thought/belief is long before statements. — creativesoul

Agreed. However, as long as thought/belief has not been expressed in language, it cannot be communicated unambiguously. We still use lots of body language, but probably not in the context of philosophy.

I don't see the similarity. — creativesoul

Gettier concludes that JTB (Justified True Belief) must be JB (Justified Belief). I conclude the same conclusion (JTB --> JB) but for different reasons.

In the debate, Is "justified true belief" a good definition for knowledge?, I agree with the answer: SImply put "Justified belief" is enough As stated so clearly by Gettier.

The correspondence theory of truth requires some kind of legitimate isomorphism between the true proposition and the history of the universe. Furthermore, I completely agree with Bertrand Russell that the alternative, i.e. the coherence theory of truth, is nonsensical.

With "appearing in the real world", I meant structurally isomorphic with the real world, in line with Bertrand Russell's considerations on the matter.

There are other reasons why knowledge is a justified belief and not necessarily a justified true belief.

For example, a mathematics theorem is a conclusion that necessarily follows from the explicit construction logic of an abstract, Platonic world, i.e. its set of axioms. If "true" means that a proposition appears in the real, physical world, then not one theorem in mathematics is "true". A provable mathematics theorem is still entirely valid knowledge.

Therefore, the JTB definition for knowledge is wrong. It must be JB instead.

This is the same conclusion as Gettier's, but obtained in a different way.

> If we assume that philosophers do create new knowledge

Philosophers rather discover new questions.

Success means that the question can actually be answered in one of the epistemically-restricted subsets of philosophy (math, science, ...). An answer that stays within general philosophy, is usually disappointing in terms of justification.

> If we formulate existence as a property of objects, then we must either admit that all objects exist

In the real world, you cannot visit all objects. You do not have enough energy for that. You can only visit a sample.

"All X", i.e. ∀x can only be visited in abstract, Platonic world, and not in the real, physical world.

You can visit (by induction) all integers, or all odd numbers, or something like that; because it is does not require energy to visit them in their abstract, Platonic world.

"Existence" is about the real, physical world.

Enumerating "all" physical objects in the real world, is simply not allowed. It is never done anyway. Experimental testing (empirical investigation) is about visiting just a few of these objects.

It is strictly forbidden to use the universal quantifier ∀ in the realm of science or any other empirical domain. You may only use it in an abstract, Platonic domain.

(all->Platonic) objects (exist->empirical) => forbidden mixing of domains

@For me, philosophy is an ongoing discussion over the nature of being.

S="What I am holding now in my hand is an apple."
Is S philosophy?