You are assuming the existence of an inherent order that lies beyond conscious recognition. Is there another aspect of mind that might register this phenomena? Is the fact we can discuss IO due to this possibility? — jgill

I don't think I'd call it an aspect of mind, rather an aspect of life. It is evident, that at the most fundamental level, living beings make use of inherent order, by creating extremely complex molecules, etc.. I don't know how the inherent order is recognized at this level, but it must be, in some way, in order for this organizational activity to occur.

I wouldn't say it is an aspect of "mind" that recognizes it because then we get into panpsychism, or something like that, and I think that there is a clear separation, or difference between mind, along with consciousness, and what happens at this fundamental level. So I think it is good to keep them separate, and understand that this is not a part of mind though mind may be a part of this. But the part only performs its particular function, without understanding its relation to the whole.

It is evident, that at the most fundamental level, living beings make use of inherent order, by creating extremely complex molecules, etc.. — Metaphysician Undercover

I brought this up earlier, the notion that "ordering" and "order" were not the same, saying that order had to do with biological and other systems, rather than listing. How order relates to ordering the elements of a set is unclear.

Yes, I think that's the distinction I've been describing to Luke, the difference between inherent order, and order as created by the human mind, what you call "ordering". The two are completely different, and how they relate is in some respects "unclear".

The issue I brought up with fishfry is the distinction between representation and imagination. Fishfry allows that "abstraction" might encompass both of these, such that imaginary ideas could be a useful part of a representative model. But this was not borne out by fishfry's map analogy. The purely imaginary in this case, is the thing with no inherent order, as a "set". Another example of the purely imaginary is infinity.

The point that I am making, is that the current order which a thing has, the inherent order, limits the possibility for future order. Therefore the possibilities for ordering (by the mind), in any true sense, must be limited by an assumed inherent order that a thing has. To remove the necessity of assuming an inherent order, for the sake of allowing infinite possibilities of ordering, is a fiction of the imagination, which has no place in a representative model.

So here's an example. Suppose there's a set which contains the "numbers" 3,4,5. The "numbers" are assumed to be abstract objects. As abstract objects, they necessarily have meaning. The meaning which the abstract objects have, necessarily gives them an order, an order which inheres within that group of objects, the set. You cannot have five without first having four, and you cannot have four without first having three. Or even if we define purely by quantity, there inheres within the meaning of "quantity", more and less. Therefore there is an order which inheres within that set of abstract objects, necessitated by the meaning of the objects. To assume to be able to remove that order, which is essential to the existence of those abstract objects, and therefore inheres with the set, for the sake of claiming that any ordering of the set is possible, is a mistaken adventure.

There is one thing about the relation between inherent order, and ordering, which is very clear. This is that to represent the possible orderings of any group of objects, one must have respect for the limitations imposed by the order that they already have. To deny that they have any inherent order is to produce a false representation based in imaginary fiction. And to give abstract objects a special status as fishfry does, to allow exemption from this rule, is to render them free from the restrictions of definition which enforces logical relations (order) between them, thus allowing the imagination to run free.

Therefore there is an order which inheres within that set of abstract objects, necessitated by the meaning of the objects — Metaphysician Undercover

But as simple symbols, rather than meaningful symbols, they may have no IO or a different IO. If I make up three random symbols from finite lines, say, would you then state the order in which I created them gives IO to the set?

But as simple symbols, rather than meaningful symbols, they may have no IO or a different IO. If I make up three random symbols from finite lines, say, would you then state the order in which I created them gives IO to the set? — jgill

You know it's an oxymoron to talk of a symbol without meaning. If it doesn't symbolize something we can't call it a symbol. Anyway, a physical symbol is not much different from any other physical thing, except it's produced with intention. Each symbol as a distinct entity has order inherent within as the relations of its parts, and as a group of three, there is order inherent in the relations between them, As symbols, with meaning, there is a reason for the order in which you write them. if they are supposed to be devoid of meaning, then why would you be making them in the first place? It's an intentional act, so there must be reasons, therefore order.

You know it's an oxymoron to talk of a symbol without meaning — Metaphysician Undercover

True enough. I should have said random configuration of line segments or something like that.

It's an intentional act, so there must be reasons, therefore order. — Metaphysician Undercover

Are all acts founded on reason?

Is there an axiom in set theory that requires the display of elements of sets to be done in inherent order?

Are all acts founded on reason? — jgill

No, not all acts are based in reason. Reason is a property of the conscious mind. But that they are not based in reason does not mean that there is not order to them, as there is order in the actions of the inanimate world. That's the difference I referred to. If all order is derived from consciousness, or "reason", we are lead toward panpsychism.

Is there an axiom in set theory that requires the display of elements of sets to be done in inherent order? — jgill

No I really do not believe there is such an axiom, as fishfry stated, a set has no inherent order. And regardless of the displayed order, one could express a statement like 'these elements without any order'. The issue is whether it is possible to conceive of something like that, 'elements without any order'. If not, it would be an incoherent concept. It is possible to state things which are contradictory, or something like that, making what is stated impossible to conceive.

Demanding that the notion of quantity is synonymous with the notion of order is misguided, for a set is normally specified as a collection of things which satisfy a given predicate, and the "quantity" of such things usually makes no reference to a constructive ordering of the elements concerned but merely to the existence of other sets for which there exists a proveable bijection to the present set, a bijection that can merely involve a bi-directional translation of any given element of a set into another.

Ironically, it is the the platonists who insist that every set must be "well-ordered" which is an assumption equivalent to the axiom of choice. But for those who deny the axiom of choice, it is nevertheless meaningful to compare the "sizes" of different sets even if the determined sizes are not synonymous to counting elements.

Then there is the little matter of potential infinity. Mathematically, it might well be the case that the number of grains of sand in a heap is neither finite nor actually infinite, but indefinitely large. To argue differently is to argue the religion of physics rather than maths.

Suppose that a heap of sand is indefinitely large, in that every time a grain of sand is extracted from the heap it might be possible to remove from the heap yet another grain of sand. Even though the heap of sand is indefinitely large, it is nevertheless meaningful to speak of the original heap of sand as being larger than the heap with a grain of sand removed, and yet in this case it is only possible to count the grains removed from the heap.

Mathematically, it might well be the case that the number of grains of sand in a heap is neither finite nor actually infinite, but indefinitely large — sime

As a mathematician who has dabbled for years in complex analysis - a branch of math that involves the limit process, calculus, etc. in the complex plane - I have rarely if ever interpreted infinity as a kind of "number" but as shorthand for "unbounded", which correlates to you always finding another grain of sand.

As for inherent order, I admit that writing {c,a,b}, although permissible, just doesn't seem right. I would always prefer {a,b,c}. :cool:

a set is normally specified as a collection of things which satisfy a given predicate, — sime

I foresee big trouble with this definition or my name's not Gottlob Frege. Well my name's not Gottlob Frege, but Frege proposed the same definition. Then Russell came along and said, "Ok smart guy, take the predicate to be $x \notin x$ and see what happens. You get a contradiction. Busted!

https://en.wikipedia.org/wiki/Russell%27s_paradox

In fact a set is ... well nobody knows what a set is. A set is whatever satisfies the axioms of some particular set theory, of which there are many. A thing could be a set in one set theory and not in another.

What you gave is the definition used in "high school set theory," sometimes called naive set theory. But that definition fails as soon as you examine it closely.

Ironically, it is the the platonists who insist that every set must be "well-ordered" which is an assumption equivalent to the axiom of choice. — sime

News to me. You claim that being a Platonist ixs equivalent to believing in the axiom of choice? I'd take those two things to be totally independent of one another. You could be a Platonist or not, and pro-choice or not. I don't see the connection.

But for those who deny the axiom of choice, it is nevertheless meaningful to compare the "sizes" of different sets even if the determined sizes are not synonymous to counting elements. — sime

The problem is that absent the axiom of choice, there are infinite sets that are not comparable to each other. That is, there are cardinals $x$ and $y$ such that none of $x = y$ nor $x \lt y$ nor $x \gt y$ are true.

That's actually the purpose of the axiom of choice, to make infinite sets behave. Without choice there's an infinite set that's Dedekind-finite; that is, a set that's not bijectively equivalent to any natural number 0, 1, 2, 3, ..., but that is not bijectively equivalent to any of its proper subsets.

So in fact your statement is inaccurate. If you deny the axiom of choice, it is sometimes MEANINGLESS to compare the sizes of sets.

for a set is normally specified as a collection of things which satisfy a given predicate . . . — sime

. . . nobody knows what a set is — fishfry

Don't feel badly. As a professor of math I rarely thought of sets that did not satisfy this definition. In passing I thought of Russell's paradox as quirky, requiring set theory be a little more sophisticated. Living in a state of "high school" naivety all those years has been devastating. How can I recover, now I am near the end of my days? :cry:

The issue I brought up with fishfry is the distinction between representation and imagination. Fishfry allows that "abstraction" might encompass both of these, such that imaginary ideas could be a useful part of a representative model. — Metaphysician Undercover

I was just reading about the Frege-Hilbert dispute. As I understand it, Hilbert was saying that axioms are formal things and it doesn't matter what they stand for as long as we can talk about their logical relationships such as consistency. Frege thought that the axioms are supposed to represent real things. I'm not sure if I'm summarizing this correctly but this is the feeling I got when I was reading it. That I'm taking Hilbert's side, saying that the axioms don't mean anything at all; and you are with Frege, saying that the axioms must mean whatever they are intended to mean and nothing else.

https://plato.stanford.edu/entries/frege-hilbert/

Is this what you're getting at?

You then complained about my map analogy, but I hope you will agree that you could object to my analogy without necessarily refuting my thesis. If you don't like maps, forget the maps.

ps -- I found another nice article.

https://academic.oup.com/philmat/article/13/1/61/1569375

You can skip the category theory stuff, scroll down to here:

During the second half of the nineteenth century, through a process still awaiting explanation, the community of geometers reached the conclusion that all geometries were here to stay … [T]his had all the appearance of being the first time that a community of scientists had agreed to accept in a not-merely-provisory way all the members of a set of mutually inconsistent theories about a certain domain … It was now up to philosophers … to make epistemological sense of the mathe‐maticians’ attitude toward geometry … The challenge was a difficult test for philosophers, a test which (sad to say) they all failed …

For decades professional philosophers had remained largely unmoved by the new developments, watching them from afar or not at all … As the trend toward formalism became stronger and more definite, however, some philosophers concluded that the noble science of geometry was taking too harsh a beating from its practitioners. Perhaps it was time to take a stand on their behalf. In 1899, philosophy and geometry finally stood in eyeball-to-eyeball confrontation. The issue was to determine what, exactly, was going on in the new geometry.

I take this to be a reference to the fact that geometers studied various models of geometry, such as Euclidean and non-Euclidean, and were no longer concerned with which was "true," but rather only that each model was individually consistent. And philosophers, who said math was supposed to be about truth, were not happy.

And:

What was going on, I believe, was that geometry was becoming less the science of space or space-time, and more the formal study of certain structures. Issues concerning the proper application of geometry to physics were being separated from the status of pure geometry, the branch of mathematics.1Hilbert's Grundlagen der Geometrie [1899] represents the culmination of this development, delivering a death blow to a role for intuition or perception in the practice of geometry. Although intuition or observation may be the source of axioms, it plays no role in the actual pursuit of the subject.

(my emphasis).

I think that last bolded part is the heart of our discussion. I'm saying that we can study aspects of the world by creating formal abstractions that, by design, have nothing much to do with the world; and that are studied formally, by manipulation of symbols. And that we use this process to then get insight about the world.

And you say, how can these meaningless abstractions possibly tell us anything about the world? They're meaningless, they're not true. You are taking the Fregean position, that if the axioms are not about the world, they're nonsense.

Here's Paul Bernays explaining Hilbert's point of view:

A main feature of Hilbert's axiomatization of geometry is that the axiomatic method is presented and practiced in the spirit of the abstract conception of mathematics that arose at the end of the nineteenth century and which has generally been adopted in modern mathematics. It consists in abstracting from the intuitive meaning of the terms … and in understanding the assertions (theorems) of the axiomatized theory in a hypothetical sense, that is, as holding true for any interpretation … for which the axioms are satisfied. Thus, an axiom system is regarded not as a system of statements about a subject matter but as a system of conditions for what might be called a relational structure … [On] this conception of axiomatics, … logical reasoning on the basis of the axioms is used not merely as a means of assisting intuition in the study of spatial figures; rather logical dependencies are considered for their own sake, and it is insisted that in reasoning we should rely only on those properties of a figure that either are explicitly assumed or follow logically from the assumptions and axioms.

(my emphasis again)

I do believe this is what our conversation is about. You're a Fregean and I'm with Hilbert.

What do you think? For my part these articles have given me some insight into your point of view. If collections in the real world have inherent order, what sense does it make to postulate sets that have no inherent order? The answer is Hilbert's side of the debate; which, for better or worse, is the prevalent view in modern math.

There's more. This is a great article.

At first, Frege had trouble with this orientation to mathematics. In a letter dated December 27, 1899, he lectured Hilbert on the nature of definitions and axioms.3 According to Frege, axioms should express truths; definitions should give the meanings and fix the denotations of terms.

(my emphasis)

This is EXACTLY what you are lecturing me about! But as SEP notes, Hilbert stopped replying to Frege in 1900. Just as I ultimately had to stop replying to you. If you and Frege don't get the method of abstraction, Hilbert and I can only spend so much time listening to your complaints. I found a great sense of familiarity in reading about the Frege-Hilbert debate.

Excellent commentary. Sitting in the bleachers awaiting MU's reply.

Excellent commentary. Sitting in the bleachers awaiting MU's reply. — jgill

I'll join you for a hot dog and a beer.

Yes, I'm already aware of all of that, and was only speaking approximately on set theory. My point was only attacking the idea that quantity is reducible to ordering.

News to me. You claim that being a Platonist ixs equivalent to believing in the axiom of choice? I'd take those two things to be totally independent of one another. You could be a Platonist or not, and pro-choice or not. I don't see the connection. — fishfry

The connection is the fact that the axiom of choice is equivalent to the law of excluded middle, which for infinite objects dissociates truth from derivation. This in itself wouldn't imply platonism if it wasn't for the fact that most proponents of classical logic and ZFC make no attempt to justify the formalisms pragmatically with respect to real world application.

The connection is the fact that the axiom of choice is equivalent to the law of excluded middle, — sime

"And of course, we know that LEM does not imply AC, since we know that ZF is consistent with ¬AC while LEM holds." (MathStackExchange) :chin:

deleted post

the axiom of choice is equivalent to the law of excluded middle — sime

That is not correct. It is the case that Z (even without the law of excluded middle (LEM)) and the axiom of choice (AC) together imply LEM. But it is not the case that Z (which includes LEM) implies AC.

"And of course, we know that LEM does not imply AC, since we know that ZF is consistent with ¬AC while LEM holds." (MathStackExchange) :chin: — jgill

Sorry for the confusion. Yes that is true for ZF, since it is built upon classical logic. In set theory, controversial instances of the excluded middle are the result of both the underlying logic if it is classical as well as the set theoretic axioms of choice and regularity.

What i had in mind wasn't ZF, but intuitionistic set theory, in which choice principles and LEM are approximately equivalent as documented in the SEP article on the axiom of choice.

https://plato.stanford.edu/entries/axiom-choice/

This stuff just gets more bewildering as time goes on. :worry:

(One reason math has become so abstract is that classical areas of investigation have been "mined out". Professors need suggestions for research topics for their PhD students. So, create new definitions and/or generalize.)

that is true for ZF, since it is built upon classical logic — sime

That is not the reason. The reason is that LEM does not imply AC, whether with intuitionistic or classical logic.

intuitionistic set theory, in which choice principles and LEM are approximately equivalent as documented in the SEP article on the axiom of choice. — sime

I looked at that article briefly. I did not see mention of an "approximate" equivalence.

Whether classically or intuitionistically, there is not an equivalence. Rather, there is only the one direction: AC implies LEM. But the other direction that is needed for equivalence - LEM implies AC - is not the case.

Explicitly constructive mathematics goes back at least a hundred years, and with roots in the 19th century too. It has great importance toward understanding foundations. I think interest in it goes well beyond any need for assigning research topics.

The connection is the fact that the axiom of choice is equivalent to the law of excluded middle, which for infinite objects dissociates truth from derivation. This in itself wouldn't imply platonism if it wasn't for the fact that most proponents of classical logic and ZFC make no attempt to justify the formalisms pragmatically with respect to real world application. — sime

I do not believe you have correctly stated Diaconescu's theorem. (Didn't think I knew that one, did you!) I quote from Wiki:

In mathematical logic, Diaconescu's theorem, or the Goodman–Myhill theorem, states that the full axiom of choice is sufficient to derive the law of the excluded middle, or restricted forms of it, in constructive set theory.

So AT BEST, AC implies LEM. And NOT the converse. So your claim of equivalence is not supported by Wikipedia. I have to trust Wiki on this point because I haven't studied this much, just remember hearing about it.

Secondly, the qualifier, "in constructive set theory." Does that mean that the implication ONLY holds in constructive set theory? Or is that an ambiguous statement on the part of some Wiki author. I don't know. But you've gone way too far in your claim of equivalence.

Moreover, you have not even remotely justified your claim that AC is equivalent to Platonism. You wrote a sentence that I could neither parse nor understand, and I don't think it's true. I'm talking about, "This in itself wouldn't imply platonism if it wasn't for the fact that most proponents of classical logic and ZFC make no attempt to justify the formalisms pragmatically with respect to real world application." Can you rephrase that so I can understand what you're saying?

(edit) I'm beginning to unpack a little of that. "Most proponents of classical logic and ZFC make no attempt to justify the formalisms pragmatically." Why on earth should they? That's not their job. And that's not even Platonism. Platonism doesn't say that the abstract math thingies refer to the real world. It says that the abstract math thingies exist in some nonphysical Platonic world. So even here I don't think you're using the technical terms correctly.

Also, "... which for infinite objects dissociates truth from derivation. " What? if you reject LEM then truth = derivation? That's false. Derivation is syntactic, truth is what's true in some model. That doesn't change just because you reject LEM.

Yes, I'm already aware of all of that, and was only speaking approximately on set theory. My point was only attacking the idea that quantity is reducible to ordering. — sime

My most humble apologies, I do not know what this is in reference to. Quantity and order are surely different, that's why there are cardinals and ordinals.

choice principles and LEM are approximately equivalent — sime

"Approximately equivalent?" What does that mean? Are 12 and 14 approximately equivalent? Apples and rutabagas? I don't think you are making your point. And to be fair I don't know much about constructive math (and not for lack of trying), so I assume you have something interesting to say here, but you're not saying it clearly.

I looked at that article briefly. I did not see mention of an "approximate" equivalence. — TonesInDeepFreeze

Glad you said that, I thought I was being too snarky jumping on that phrase.

"And of course, we know that LEM does not imply AC, since we know that ZF is consistent with ¬AC while LEM holds." (MathStackExchange) — jgill

@jgill This place is corrupting you! You're supposed to say that as a complex variables guy you don't think much about foundations! This was most impressive.

This stuff just gets more bewildering as time goes on. :worry:

(One reason math has become so abstract is that classical areas of investigation have been "mined out". Professors need suggestions for research topics for their PhD students. So, create new definitions and/or generalize.) — jgill

Constructive math is making a comeback these days because of the influence of computers. There are computerized proof assistants, and they tend to operate on some form of constructive logic or intuitionistic type theory. Brouwer was the great 1930's intuitionist, so I thin of all this as "Brouwer's revenge."

deleted post — GrandMinnow

Would that others show so much circumspection. Myself included from time to time.

Observe that the meaning of choice principles are different in constructive logic than in classical logic, and recall that the controversies over LEM and AC concern only their implied non-constructive content.

Bear in mind

1) All of the non-constructive content of classical logic is discarded by jettisoning LEM.

2) The axiom of choice holds trivially as a tautology in sets constructed in higher-order constructive logic, because in this logic existence is synonymous with construction.

So one could even say that absence of LEM implies AC (or perhaps rather, that AC is an admissible tautology in absence of LEM).

But this statement isn't enlightening, because it conflates the difference in meaning that AC has in the two different systems, for AC holds trivially and non-controversially in constructive logic as a tautology, where it doesn't imply anything above and beyond construction.

In the constructive sense, i think it is fair to say that LEM implies AC, when speaking of AC not in the sense of an isolated axiom, but in the commonly used informal vernacular when speaking of choice principles in their structural and implicational senses

the difference in meaning that AC has in the two different systems — sime

Of course, semantics for intuitionistic systems are different from semantics for classical systems. But the question of equivalence is that of derivability.

The axiom of choice holds trivially as a tautology in sets constructed in higher-order constructive logic — sime

Reference please.

So one could even say that absence of LEM implies AC (or perhaps rather, that AC is an admissible tautology in absence of LEM). — sime

That makes no sense. 'Implies' means proves in this context. And one cannot prove what is otherwise not provable by weakening the proof logic. (But I can't opine on 'admissible tautology' since I don't know your definition.)

when speaking of AC not in the sense of an isolated axiom, but in the commonly used informal vernacular when speaking of choice principles in their structural and implicational senses — sime

AC is an exact formulation. It is not expressed as "commonly used informal vernacular when speaking of choice principles in their structural and implicational senses", whatever you might exactly mean by that.

There are different formulations that may have equivalences, and there are complications throughout, but I know of no proof nor mention in the article you cited that shows the equivalence of AC with LEM in intuitionistic set theory. The SEP article does say "each of a number of intuitionistically invalid logical principles, including the law of excluded middle, is equivalent (in intuitionistic set theory) to a suitably weakened [italics in Bell's earlier article] version of the axiom of choice. Accordingly these logical principles may be viewed as choice principles." But the question was not that of various choice principles but of AC itself, and we have not been shown a proof that AC and LEM are equivalent in intuitionistic set theory.

Explicitly constructive mathematics goes back at least a hundred years, and with roots in the 19th century too. It has great importance toward understanding foundations. I think interest in it goes well beyond any need for assigning research topics. — TonesInDeepFreeze

My comment about abstraction wasn't really in reference to foundations, just a general reflection on the profession. I mean no disrespect to constructive mathematics. I recall my advisor fifty years ago talking about classical analysis being somewhat mined out and mathematicians moving toward generalities and abstraction to find new territory. He thought interest in the classical might come back at some point.

(Actually, I'm a fan of Brouwer, more so of Banach, whose fixed point theorem I extended to suit my special interests.)

jgill This place is corrupting you! — fishfry

Makes my head spin. Thanks for opening my eyes a bit to formalized set theory and for your patience!

I mean no disrespect to constructive mathematics. — jgill

I disrespect constructive math all the time. Actually over the past couple of years a handful of constructivists showed up here and I had interesting conversations with them, but always ended up baffled and they left. I hope it wasn't something I said, I'm just trying to learn. These were generally people coming to the subject from the computer science side and never the math side, so they had no appreciation for standard math. Made for frustrating conversations both ways.

Thanks for opening my eyes a bit to formalized set theory and for your patience! — jgill

Glad to help.

I was just reading about the Frege-Hilbert dispute. As I understand it, Hilbert was saying that axioms are formal things and it doesn't matter what they stand for as long as we can talk about their logical relationships such as consistency. Frege thought that the axioms are supposed to represent real things. I'm not sure if I'm summarizing this correctly but this is the feeling I got when I was reading it. That I'm taking Hilbert's side, saying that the axioms don't mean anything at all; and you are with Frege, saying that the axioms must mean whatever they are intended to mean and nothing else.

https://plato.stanford.edu/entries/frege-hilbert/

Is this what you're getting at? — fishfry

Thanks for the reading material fishfry, I've read through the SEP article a couple times, and the other partially, and finally have time to get back to you.

I think Frege brings up similar issues to me. The main problem, relevant to what I'm arguing, mentioned in the referred SEP article, is the matter of content.

The difference of opinion over the success of Hilbert’s consistency and independence proofs is, as detailed below, the result of significant differences of opinion over such fundamental issues as: how to understand the content of a mathematical theory, what a successful axiomatization consists in, what the “truths” of a mathematical theory really are, and finally, what one is really asking when one asks about the consistency of a set of axioms or the independence of a given mathematical statement from others. — SEP

In critical analysis, we have the classical distinction of form and content. You can find very good examples of this usage in early Marx. Content is the various ideas themselves, which make up the piece, and form is the way the author relates the ideas to create an overall structured unity.

Hilbert appears to be claiming to remove content from logic, to create a formal structure without content. In my opinion, this is a misguided adventure, because it is actually not possible to pull it off, in reality. This is because of the nature of human thought, logic, and reality. Traditionally, content was the individual ideas, signified by words, which are brought together related to each other, through a formal structure. Under Hilbert's proposal, the only remaining idea is an ideal, the goal of a unified formal structure. So the "idea' has been moved from the bottom, as content, to the top, as goal, or end. This does not rid us of content though, as the content is now the relations between the words, and the form is now a final cause, as the ideal, the goal of a unified formal structure. The structure still has content, the described relations.

Following the Aristotelian principles of matter/form, content is a sort of matter, subject-matter, hence for Marx, ideas, as content, are the material aspect of any logical work. This underlies Marxist materialism

However, in the Aristotelian system, matter is fundamentally indeterminate, making it in some sense unintelligible, producing uncertainty. Matter is given the position of violating the LEM, by Aristotle, as potential is is what may or may not be. Some modern materialists, dialectical materialists, following Marx's interpretation of Hegel prefer a violation of the law of non-contradiction.

So the move toward formalism by Frege and Hilbert can be seen as an effort to deal with the uncertainty of content, Uncertainty is how the human being approaches content, as a sort of matter, there is a fundamental unintelligibility to it. Hilbert appears to be claiming to remove content from logic, to create a formal structure without content, thus improving certainty. In my opinion, what he has actually done is made content an inherent part of the formalized structure, thus bringing the indeterminacy and unintelligibility, which is fundamental to content, into the formal structure. The result is a formalism with inherent uncertainty.

I believe that this is the inevitable result of such an attempt. The reality is that there is a degree of uncertainty in any human expression. Traditionally, the effort was made to maintain a high degree of certainty within the formal aspects of logic, and relegate the uncertain aspects to a special category, as content.

Think of the classical distinction between the truth of premises, and the validity of the logic. We can know the validity of the logic with a high degree of certainty, that is the formal aspect. But the premises (or definitions, as argued by Frege) contain the content, the material element where indeterminateness, unintelligibility and incoherency may lurk underneath. We haven't got the same type of criteria to judge truth or falsity of premises, that we have to judge the validity of the logic. There is a much higher degree of uncertainty in our judgement of truth of premises, than there is of the validity of logic. So we separate the premises to be judged in a different way, a different system of criteria, knowing that uncertainty and unsoundness creeps into the logical procedures from this source.

Now, imagine that we remove this separation, between the truth of the premises, and the validity of the logic, because we want every part of the logical procedure to have the higher degree of certainty as valid logic has. However, the reality of the world is such that we cannot remove the uncertainty which lurks within human ideas, and thought. All we can do is create a formalism which lowers itself, to allow within it, the uncertainties which were formerly excluded, and relegated to content. Therefore we do not get rid of the uncertainty, we just incapacitate our ability to know where it lies, by allowing it to be scattered throughout the formal structure, hiding in various places, rather than being restricted to a particular aspect, the content.

I will not address directly, Hilbert's technique, described in the SEP article as his conceptualization of independence and consistency, unless I read primary sources from both Hilbert and Frege.

I take this to be a reference to the fact that geometers studied various models of geometry, such as Euclidean and non-Euclidean, and were no longer concerned with which was "true," but rather only that each model was individually consistent. And philosophers, who said math was supposed to be about truth, were not happy. — fishfry

Geometers, and mathematicians have taken a turn away from accepted philosophical principles. This I tried to describe to you in relation to the law of identity. So there is no doubt, that there is a division between the two. Take a look at the Wikipedia entry on "axiom" for example. Unlike mathematics, in philosophy an axiom is a self-evident truth. Principles in philosophy are grounded in ontology, but mathematics has turned away from this. One might try to argue that it's just a different ontology, but this is not true. There is simply a lack of ontology in mathematics, as evidenced by a lack of coherent and consistent ontological principles.

You might think that this is all good, that mathematics goes off in all sorts of different directions none of which is grounded in a solid ontology, but I don't see how that could be the case.

I'm saying that we can study aspects of the world by creating formal abstractions that, by design, have nothing much to do with the world; and that are studied formally, by manipulation of symbols. And that we use this process to then get insight about the world. — fishfry

I know you keep saying this, but you've provided no evidence, or proof. Suppose we want to say something insightful about the world. So we start with what you call a "formal abstraction", something produced from imagination, which has absolutely nothing to do with the world. Imagine the nature of such a statement, something which has nothing to do with the world. How do you propose that we can use this to say something about the world. It doesn't make any sense. Logic cannot proceed that way, there must be something which relates the abstraction to the world. But then we cannot say that the abstraction says nothing about the world. If the abstraction is in some way related to the world, it says something about the world. If it doesn't say anything about the world, then it's completely independent from any descriptions of the world, so how would we bring it into a system which is saying something about the world?

This is EXACTLY what you are lecturing me about! But as SEP notes, Hilbert stopped replying to Frege in 1900. Just as I ultimately had to stop replying to you. If you and Frege don't get the method of abstraction, Hilbert and I can only spend so much time listening to your complaints. I found a great sense of familiarity in reading about the Frege-Hilbert debate. — fishfry

Actually, I think that often when one stops replying to the other it is because they get an inkling that the other is right. So there's a matter of pride, where the person stops replying, and sticks to one's principles rather than going down the road of dismantling what one has already put a lot of work into, being too proud to face that prospect. You, it appears, do not suffer from this issue of pride so much, because you keep coming back, and looking further and further into the issues.

it is not the case that Frege and I do not get the method of abstraction. Being philosophers, we get abstraction very well, it is the subject matter of our discipline. You do not seem to have respect for this. This is the division of the upper realm of knowledge Plato described in The Republic. Mathematicians work with abstractions that is the lower part of the upper division, philosophers study and seek to understand the nature of abstractions, that is the upper half of the upper division. Really, it is people like you, who want to predicate to abstraction, some sort of idealized perfection, where it is free from the deprivations of the world in which us human beings, and our abstractions exist, who don't get abstraction.

And then there are those occasional "aha!" moments when a physicist uses a mathematician's abstractions to describe a physical process well enough that accurate predictions ensue. Accidental, or subconsciously generated to match physical reality?