A couple of months ago the forum was infested with bad theology. Now it's bad maths. — Banno

So, we're making progress. :strong:

Guess so.

Besides, one shouldn't assume that one school of Mathematical philosophy is correct and another is not. — ssu

And you still haven't grasped the very simple fact that no field of mathematics claims to be "correct", or that another is not. Only that no statement is can be shown to be true without first assuming a set of unsupported Axiom, and proving theorems within that framework.

And it is quite clear that you have no interest in any formalism but your own.

That does not mean that every "proposition regarded as self-evidently true without proof" is an axiom. — JeffJo

Perhaps you can point to an example of a "proposition regarded as self-evidently true without proof' that is not an axiom?

So once again, the statement you claimed was an axiom was never stated as part of such a set, from which theorems could be derived. It was not an axiom, it was a near-religious belief. — JeffJo

Yes it is, the axiom of infinity is part of the Zermelo–Fraenkel system of axioms:

"Together with the axiom of choice (see below), these are the de facto standard axioms for contemporary mathematics or set theory."

https://en.wikipedia.org/wiki/List_of_axioms#ZF_(the_Zermelo–Fraenkel_axioms_without_the_axiom_of_choice)

No. What I am saying is that theorems in a field of mathematics need to be based on some set of accepted truths that are called the axioms of that field. Such a set can be demonstrated to be invalid as a set by deriving a contradiction from then, but not by comparing them to other so-called "truths" that you choose to call "self-evident." — JeffJo

You are confused. Perhaps another example will help:

Special relativity is derived from two axioms:

1. The laws of physics are the same in all inertial frames of reference
2. The speed of light in a vacuum is the same for all observers, regardless of the motion

If either of these axioms is ever demonstrated (by experiment or a theoretical argument) not to be true, then all the results of special relativity will be invalidated.

Any 'mathematical truth' is only as good as its axioms - if any of the axioms are false, everything derived from that axiom is invalidated.

Sorry, I was rude. Let me give it another try.

An axiom is a proposition regarded as self-evidently true without proof. — ssu

This is an antiquated definition, suitable perhaps as an informal introduction to the topic, but not suitable for today's mathematics. And it's not about formalism vs. intuitionism or whatnot. For one thing, this formulation just isn't accurate and doesn't capture the role of axioms, even in Euclid's original books. For example, the fact that a square with the side equal to 1 cannot be inscribed inside a circle with the radius equal to 1 may be self-evidently true, but Euclid did not make it an axiom. Axioms are those propositions that are specifically chosen as the primitive building blocks of a mathematical theory.

More importantly, if axioms were a matter of self-evident truths, then there would be just the one mathematics, because there is only one truth (at least that's how most people see it). But this hasn't been the case with mathematics since long before people even started contemplating foundational philosophical questions like formalism, logicism, etc. The notion that mathematical axioms are some extra-mathematical truths (truths about what?) has been abandoned.

And you still haven't grasped the very simple fact that no field of mathematics claims to be "correct", or that another is not. Only that no statement is can be shown to be true without first assuming a set of unsupported Axiom, and proving theorems within that framework.

And it is quite clear that you have no interest in any formalism but your own. — JeffJo

Perhaps you didn't understand my point.

I'm not looking for some ultimate truth. The question is if a set of axioms, an axiomatic system, is simply consistent. If they aren't consistent, I would in my mind declare then an axiom or axioms to be false, if we can pinpoint the reason for the inconsistency. This kind of "formalism" I do accept. So the question of the possibility of "an axiom being false" would perhaps be better understood from your viewpoint as that "an axiomatic system is inconsistent".

Could that be possible? Let's take the example of the axioms of ZF set theory. Consider the reason just why Zermelo and Fraenkel made those axioms: it was because of Russell's Paradox, which had made Frege's naive set theory, well...."naive". There was a reason why to do it. Yet perhaps the Paradoxes aren't at all an obstacle to be eradicated, but simply part of an answer we haven't fully understood. Because we don't understand infinity clearly, there's still CH you know, our understanding of these issues can change. Then axioms denying Paradoxes (assuming if they would be a part a reductio ad absurdum proof) would be, well, some might say false, others would say that the system wouldn't be consistent.

You might argue that fine, that doesn't matter, lets just form a new Set Theory and leave ZF as it is. But if so, are you OK with an axiomatic system where 0=1? In that system you can prove truly whatever you want! I might argue that the 0=1 is false, but I do understand your formalist point of view. Perhaps you would get angry at me saying that, because for you it's just an axiomatic system as anything else. If you hate the true/false dichotomy and juxtaposition, how about useful / useless then? If we have one axiomatic system, which is very useful to us, we can model extremely many things with it and another that cannot be used in any way, is there something to be said about the axioms.

I think that these questions go to the core of the philosophy of Mathematics.

I think I understand your point. Perhaps my answer to JeffJo above will make my point more clear. It's better to think of axioms as part of axiomatic systems. Yet when it comes to mathematics, is every axiomatic system as useful as the other? We tend to use some systems more than others, at least.

More importantly, if axioms were a matter of self-evident truths, then there would be just the one mathematics — SophistiCat

What would be so terrible if it would be so? Now it isn't, I agree with that wholeheartedly, but just making a hypothesis here. Could there be an universal foundation for Mathematics?

Your intuition is constructively valid, since constructivists identify the real reals with the computable binary sequences, that is to say, the binary sequences that are computable total functions, and since they reject the premise of Cantor's theorem that the power-set of the natural numbers exists (due to most of it's elements being non-computable), they instead regard diagonal arguments as merely showing that a hypothetical enumeration of only the genuine real numbers cannot be formulated.

Unfortunately in Cantors day there wasn't much attention paid to the underlying algorithms used to generate binary sequences, and hence he failed to admit the critical distinctions between

-The Total functions corresponding to the countable set of reals that can actually be constructed by an algorithm that halts to produce a digit for every argument representing a position of the sequence.

-The Provably Total functions corresponding to the subset of constructive reals that are 'a priori' deducible, in the sense that their underlying algorithms can be proved to halt without requiring the algorithms in question to be actually run.

-The Non-Provably Total functions corresponding to the subset of 'a posteriori' deducible constructable reals , whose algorithms in fact halt when run, but whose halting cannot be proved without running the algorithms in question.

-The Partial functions that fail to produce digits for every position of the sequence, and hence fail to represent a legitimate number, a property which is generally unknowable.

Cantor's diagonal argument when applied to the countable set of provably total functions, constructs a 'diagonal' total function, i.e. an additional valid real number, that is no longer provably total, and we also know that the combined set of provably total and non-provably total functions is countable (via enumerating their respective Turing machines).
Yet we cannot apply a diagonal argument to the set of non-provably total functions to produce an additional real number, for their very non-provability forbids us from knowing apriori whether or not our list contains only total functions representing genuine real numbers. Therefore there is no constructively acceptable diagonal method for producing a new real number from an enumeration of all and only the real numbers.

Most working mathematicians do not feel mortally obliged to prove the existence of an object by producing the object. It is best if they can, however. If they can show that assuming the non-existence of the object infers a fundamental violation of mathematical reason - an indirect proof - that suffices. The Law of the Excluded Middle was quite useful in topology (sometimes referred to as math without numbers, with which I disagree).

I assume all of you grok the sophisticated presentations on this thread. No need to worry so much about all these technical details! Life will go on.

You guys are so HARD on Cantor! Are you not aware that normal, everyday mathematics produces excellent results? And much of that, done on a computer, treats all numbers as rational. Are you really that concerned with non-computable functions or non-measurable sets? Material like that in math is referred to as "pathological" frequently. :meh:

A couple of months ago the forum was infested with bad theology. Now it's bad maths. — Banno

In mathematics, instrumentalism reigns supreme. Anyone who rejects instrumentalism, opting for truth as a first principle, is accused of "bad maths". However, the terms of "bad" and "good" receive their meaning from morality. So to resolve the issue of whether it is really the instrumentalist, or the truth seeker who has "bad maths", we would need to apply moral principles.

Do you have any moral principles which show that instrumentalism is better than truth seeking?

re you really that concerned with non-computable functions or non-measurable sets? Material like that in math is referred to as "pathological" frequently. — John Gill

We ought to treat the existence of non-computability and incommeasurability much more seriously than we do. Yet mathematicians push them aside and think somehow that they are 'negative' or something that ought to be avoided.

I personally think that absolutely everything is mathematical or can be described mathematically. Huge part is just non-computable. When we would understand just what is non-computable, we would avoid banging our heads into the wall with assuming that everything would be computable.

Perhaps you didn't understand my point. — ssu

It's clear you don't understand mine. Nor have you tried.

The question is if a set of axioms, an axiomatic system, is simply consistent.

Yes, it is. That is exactly what you have not addressed.

If they aren't consistent, I would in my mind declare then an axiom or axioms to be false

And you would be wrong to do so. All it shows is that the set is inconsistent. Any of the axioms could individually be part of a different, consistent, set. Yet you are calling an axiom, or axioms, "false" in a sense that can only be called "ultimate" or "absolute."

Get this point straight: The Axiom of Infinity cannot be proven to be true, or false, outside of some set of Axioms. I believe your words were that that his discussion should establish whether the AoI is self-evidently true. Nothing is further from the point if this discussion.

Given a consistent set theory that accepts the existence of an infinite set, we require different sizes of such sets. And not believing in infinity cannot change that.

We ought to treat the existence of non-computability and incommeasurability much more seriously than we do. — ssu

Wiki: incommensurable generally refers to things that are unlike and incompatible, sharing no common ground (as in the "incommensurable theories" of the first example sentence), or to things that are very disproportionate, often to the point of defying comparison ("incommensurable crimes"). Both words entered English in the 1500s and were originally used (as they still can be) for numbers that have or don't have a common divisor.

Not quite the same as mathematical measure theory. But the above may have more relevance to the thread.

Wiki: According to the Church–Turing thesis, computable functions are exactly the functions that can be calculated using a mechanical calculation device given unlimited amounts of time and storage space. Equivalently, this thesis states that a function is computable if and only if it has an algorithm.
Every computable function has a finite procedure giving explicit, unambiguous instructions on how to compute it. Furthermore, this procedure has to be encoded in the finite alphabet used by the computational model, so there are only countably many computable functions. For example, functions may be encoded using a string of bits (the alphabet Σ = {0, 1}). The real numbers are uncountable so most real numbers are not computable. See computable number. The set of finitary functions on the natural numbers is uncountable so most are not computable. Concrete examples of such functions are Busy beaver, Kolmogorov complexity, or any function that outputs the digits of a noncomputable number, such as Chaitin's constant.

It's amazing at how well computers have served us, isn't it, given these restrictions? :cool:

You guys are so HARD on Cantor! — John Gill

Just some folk are. :)

We ought to treat the existence of non-computability and incommeasurability much more seriously than we do. Yet mathematicians push them aside and think somehow that they are 'negative' or something that ought to be avoided. — ssu

There are a bunch of areas in computer science on computability and such, e.g. ...

Computational Complexity Theory (Stanford Encyclopedia of Philosophy)
Computational complexity theory, Computational complexity (Wikipedia)
NP-completeness, NP (complexity), P versus NP problem (Wikipedia)

Within some limits you can write code to handle infinite sets.
Nowhere near what mathematicians routinely do, but some things are possible.

Get this point straight: The Axiom of Infinity cannot be proven to be true, or false, outside of some set of Axioms. — JeffJo

This is a straw-man argument. Just like we cannot escape theories in other fields, we cannot escape axiomatic systems. What my point was that as we have things like CH, we don't understand Infinity yet clearly. Hence there is the possibility that for example the axioms of a axiomatic system that we think is consistent might be proven inconsistent. Just as the fate of naive set theory. I don't understand why you won't believe our understanding of math could continue to change as it has changed in history.

And you would be wrong to do so. All it shows is that the set is inconsistent. — JeffJo

No, the axioms are inconsistent to each other in the defined axiomatic system.

I believe your words were that that his discussion should establish whether the AoI is self-evidently true. Nothing is further from the point if this discussion. — JeffJo

Wrong. As I said: "I'm not looking for some ultimate truth. The question is if a set of axioms, an axiomatic system, is simply consistent. I just happen to be such a logicist that I think that something that is inconsistent in math is in other words false.

There are a bunch of areas in computer science on computability and such, e.g. ...

Computational Complexity Theory (Stanford Encyclopedia of Philosophy)
Computational complexity theory, Computational complexity (Wikipedia)
NP-completeness, NP (complexity), P versus NP problem (Wikipedia)

Within some limits you can write code to handle infinite sets.
Nowhere near what mathematicians routinely do, but some things are possible. — jorndoe

True jorndoe, in my view it's a field we likely could find something new. The Church-Turing thesis is quite vague in my view. I think the most important issue here in the most simple format is Cantor's diagonalization. It seems with logic has a lot of peculiar things happening.

This is a straw-man argument. — ssu

It is how Mathematics works. Anything that "exists" has to be based on Axioms.

we don't understand Infinity yet clearly.

Now that's a strawman argument. You need the AoI before you can even try to understand this thing you want to call "infinity."

No, the axioms are inconsistent to each other in the defined axiomatic system.

And you have proven this? Or are you just supposing it could be so?

As I said: "I'm not looking for some ultimate truth.

Yes, you did say that. You have also said that the AoI could be "wrong" and that we need to discuss whether it is.These statements contradict each other. This makes your axiomatic system inconsistent, and "false" by your definition.

The question is if a set of axioms, an axiomatic system, is simply consistent. I just happen to be such a logicist that I think that something that is inconsistent in math is in other words false.

Not ultimately false, or absolutely false, but some other kind of "false"? What kind?

Now that's a strawman argument. You need the AoI before you can even try to understand this thing you want to call "infinity." — JeffJo

Quite circular reasoning you have there, Jeffjo.

You have also said that the AoI could be "wrong" and that we need to discuss whether it is. — JeffJo

The axiom of infinity could be wrong in the way that it is inconsistent with the other axioms of ZF, for example. It is you that is making the case of some eternal truth as you don't take into consideration at all that the now used axiomatic systems could be inconsistent. I'm really not making the case for some universal truth here either. My point is that from the historical perspective we have thought about math one way and because of new theorems or observations we have changed our way of thinking about math. Why would you assume that now at this it wouldn't be so as earlier?

Not ultimately false, or absolutely false, but some other kind of "false"? What kind? — JeffJo

You tell me. All I understand is that if something is inconsistent, we can say it's false.

. . . you don't take into consideration at all that the now used axiomatic systems could be inconsistent. — ssu

All you have to do is come up with an example showing this to be the case, rather than argue in an abstract way about it. Maybe you have, as I haven't read all the posts. Good luck.

We ought to treat the existence of non-computability and incommeasurability much more seriously than we do. Yet mathematicians push them aside and think somehow that they are 'negative' or something that ought to be avoided.

I personally think that absolutely everything is mathematical or can be described mathematically. Huge part is just non-computable. When we would understand just what is non-computable, we would avoid banging our heads into the wall with assuming that everything would be computable. — ssu

Computability isn't a mathematical assumption, rather computability refers to the very activity of construction by following rules. Since mathematical logic consists only of the constructive activity of rule-following, the idea that mathematical logic can capture the non-constructive notion of "non-computability" is a contradiction in terms. None of Cantor's conclusions are really captured in his (ironically constructive) syntactical expressions. He is merely projecting his theological intuitions onto logical syntax.

Science on the other hand, attempts to predict the course of nature using a particular set of rules. Yet there is no reason to believe that the course of nature follows any particular set of rules. Hence we could say that nature might be "non-computable", but this "non-computability" cannot become the object of mathematical study on pain of contradiction.

Quite circular reasoning you have there, Jeffjo. — ssu

Do think you understand the point of Axioms? Maybe you need to explain what you think it is. Because it is your arguments that are circular.

The axiom of infinity could be wrong in the way that it is inconsistent with the other axioms of ZF, for example.

And Santa Claus could visit my house tomorrow night. But I don't draw conclusions from suppositions like that.

It is you that is making the case of some eternal truth ...

You are the one suggesting that statements could be called true, or false, outside of an axiomatic system. All I'm saying th that the AoI can be part of a consistent system, and you can't conclude anything about "Infinity" outside of one.

I'm really not making the case for some universal truth here either.

Yes, you are.

My point is that from the historical perspective we have thought about math one way and because of new theorems or observations we have changed our way of thinking about math.

No, we have not. We may have changed the Axioms.

All I understand is that if something is inconsistent, we can say it's false.

Define what "something" represents here. Because an Axiom, by itself, cannot be this "something" here yet youy keep treating it as though it can.

A ****SET***** of axioms can be inconsistent, which only means that at least one of them disagrees with one or more of the others. Not that any of them is "false." And claiming otherwise is claiming that a universal truth exists.

Since mathematical logic consists only of the constructive activity of rule-following, the idea that mathematical logic can capture the non-constructive notion of "non-computability" is a contradiction in terms. — sime

Do you understand Turing's answer to the Halting problem? Just as Cantor's diagonal argument shows that not every infinite set of numbers can be put into 1-to-1 correspondence with the Natural numbers, so do the various undecidability results, starting from Church-Turing thesis, show that indeed there are mathematical objects that cannot computed. Not everything can be calculated/computed by a Turing Machine.

All you have to do is come up with an example showing this to be the case, rather than argue in an abstract way about it. Maybe you have, as I haven't read all the posts. Good luck. — John Gill

Feel free to think that there is nothing that we could understand better in mathematics any time ever. All I said that what one could easily see even from this forum is that we do not understand infinity yet.

No, we have not. We may have changed the Axioms. — JeffJo

So changing the axioms isn't changing the way think about math?

A ****SET***** of axioms can be inconsistent, which only means that at least one of them disagrees with one or more of the others. Not that any of them is "false." And claiming otherwise is claiming that a universal truth exists. — JeffJo

Right. So are against something the idea that if something is inconsistent (in math/logic), it is false, because that would be a 'universal truth'. I guess you oppose talking about "The Law of excluded middle" because for you it's just one axiomatic system.

All I said that what one could easily see even from this forum is that we do not understand infinity yet. — ssu

It's hard to argue with that. :chin:

It's hard to argue with that. :chin: — John Gill

And I know that I may not be the sharpest razors here when it comes to math and hence I'm happy if I am shown to be wrong.

Yet I think there still is important things to be discovered in math. Just a hunch...

Do you have any moral principles which show that instrumentalism is better than truth seeking? — Metaphysician Undercover

Maths is constructed. One can do with it as one pleases with the symbols involved. We make the rules up as we go, and we can and do go back and change them as we like. They are not tied to instruments or forms or anything other than themselves.

Maths is constructed. One can do with it as one pleases with the symbols involved. We make the rules up as we go, and we can and do go back and change them as we like — Banno

It's a little more complicated than that. But go ahead and make one up. Should be fun. :smile:

OK.


Banno's Game

All I said that what one could easily see even from this forum is that we do not understand infinity yet. — ssu

You said more than that; this is just your go-to defense: to invoke the mysteriousness of infinity, like some invoke the mysteriousness of God. And yet even this seemingly innocuous banality says more than you think it does. It implies that there is some extra-mathematical Infinity that mathematics is trying to grapple with. But mathematics as such doesn't contain anything extra-mathematical. Everything in mathematics exists only to the extent to which it is defined. The Axiom of Infinity is just a name for an axiom (a family of axioms in various systems); it plays by the same rules as every other axiom and doesn't purport to refer to something extra-mathematical - unless you want it to; but that would be an extra-mathematical choice on your part, as would be any use of mathematics to model something extra-mathematical. "Infinity" in mathematics is just a name, a symbol that could be replaced with any other symbol salva veritate. There is nothing inherently pathological about it.