In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. — alcontali

Which is why a consequent of formalism is that math, as a meaningless game, is of no intrinsic value. This view is incompatible both with our experience of learning math by reflecting on examples, and with the fact that mathematical propositions are treated as truths in scientific thought.

According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other contensive subject matter — in fact, they gi't "about" anything at all. — alcontali

The Aristotelian-Thomistic view also rejects actual numbers, sets, and triangles in extramental reality, but sees an alternative other that empty formalism. Reflecting on the role of examples in learning math and on its applicability in science, it sees that numbers, sets, and triangles are intelligible (potential, able to be understood) in reality. The act of abstraction, which is one function of awareness (the agent intellect), makes what was merely intelligible in nature actually understood. This provides a middle ground between Platonism and formalism.

There may be an esoteric link between the abstract, Platonic world of mathematics and the real, physical world, but this hypothetical link cannot be used for any practical purpose. — alcontali

As there is no Platonic world, there is no possibility of a link to it, There is, however, a natural world with well-known links to mathematical thought.

This formula game enables us to express the entire thought-content of the science of mathematics in a uniform manner ... — alcontali

Your unnamed authority agrees that math is a science. Games are not sciences. Being a science (an organized body of knowledge) means that math is an understanding of reality.

Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. — alcontali

Clearly, this is nonsense, We know that there is no intrinsic necessity to the parallel postulate in the context of geometry, or to the axiom of choice in the context of Zermelo–Fraenkel set theory. If the rules are arbitrary, so are the results.

These rules form a closed system that can be discovered and definitively stated. — alcontali

No one can discover what does not pre-exist. Yet, the pre-existence of mathematical axioms is the exact premise formalists reject. So, again your authority is inconsistent.

What "truth" or objectivity can be ascribed to this theoretic construction of the world, which presses far beyond the given, is a profound philosophical problem. — alcontali

This is the problem of universals, solved by the moderate realist insight that ideogenesis involves the actualization of intelligibility in nature via abstraction by the agent intellect.

It is closely connected with the further question: what impels us to take as a basis precisely the particular axiom system developed by Hilbert? — alcontali

The answer to this is, nothing. Abstraction fixes on certain notes of intelligibility and certain intelligible relations to the exclusion of others. Thinkers may have different conceptual spaces in light of their individual experiences and needs. So, the same reality can be understood in different, partial. ways -- depending on the perspective we take and the conceptual space into which we project our experience.

Mathematics is consistent by design while the real, physical world is consistent by assumption. — alcontali

No, the physical world is consistent in virtue of its existence. The nature of being is such that it cannot instantiate a contradiction. That does not mean that hypothetical theories, in physics or in math, need be consistent.

Therefore, it is sometimes possible to construct consistency isomorphisms between both, that will be uncannily effective in mirroring some sector of reality inside an abstract, Platonic model. — alcontali

This view makes the applicability of math to nature entirely accidental. If you think about it, you'll see that you can't construct such an isomorphism unless the relevant mathematical relations are already instantiated in nature -- and we can understand that they are. But, if they are already instantiated and intelligible, both Platonism and formalism are wrong. We can construct the relevant math on the basis of our understanding of those intelligible relations.

Yes, in the Metaphysics where he discusses the principle of contradiction.

A secondary source is not a citation from Aristotle. I've read his analysis of axiomatic foundations. While he says we cannot deduce everything, he is convinced that we can justify axioms non-deductively and does so in a number of instances.

You are avoiding the question. Science does not simply "assume its principles". It determines them through observation, hypothesis, testing, theory, modeling, and so on. — Fooloso4

This is just a verbal difference. Scientists certainly do, and that it my point: axioms need justification. The verbal difference is in how to define a science like math. Some would say that no science justifies its own assumptions, others look at what people who call themselves scientists or mathematicians actually do. I don't care how you define a science such as math. We seem to be agreeing on what is and needs to be done.

First, someone has to do the abstracting. Second, the properties of say a triangle are not determined by abstraction. — Fooloso4

Yes, we need knowing/observing subjects. And, yes, abstraction does not create content, it actualizes intelligibility already present in reality.

2+2=4 does not exist only in the "Platonic realm", does not need to "apply to reality", and it is meaningless to call it a Platonic relationship. It does not apply to reality because it is counting something real. — Fooloso4

The context was that of showing the consequence of the questionable claim that 2 + 2 = 4 exists in a Platonic realm. It was not me stating my own position.

So, in a topic entitled The Foundations of Mathematics, the actual foundations of mathematics is not your present interest. — Fooloso4

My interest is how the foundations are justified. The actual foundations are of interest only as examples of claims made and needing justification. I have used a number of actual postulates in that way. Enumerating all the postulates in all branches of mathematics would not help us understand the justification processes. It would only be a distraction.

It is not simply adding new concepts, it is a matter of different concepts. — Fooloso4

I am not sure how you distinguish different concepts that were not in prior use from new concepts. Perhaps examples would help.

This does not vitiate old concepts in the sense that they are wrong, but that mathematics no longer operates according to the older concepts. — Fooloso4

I think we are using "concept" in different senses. I am thinking of <number>, <line>, <irrational number> and so on when I say "concept." You seem to be thinking rules of procedure. Yes, there are new procedures, but Euclidean geometry is still being pretty much as he wrote it in the Elements. So, the old procedures are not abandoned.

Thank you for the link/reference.

As I pointed out, there is no number 0 or 1 in Greek mathematics. You might dismiss this as simply wrong, but in doing so what you miss is the ability to understand a way of looking at the world that is not our own. — Fooloso4

No, I don't dismiss different conceptual spaces as wrong -- they are just different ways of thinking about the same reality.

Thank you. I will read it when I have time.

So in the way as history is a science? Some in the natural sciences would shudder at the idea, but I'm totally OK with it. — ssu

We seem to be converging. I see good history as the result of rigorous method, but not as explaining events from first principles.

The way I see here math to be logical that simply every mathematical truth has to be logical. It doesn't state AT ALL that everything in math has to begin from a small finite set of axioms. What Hilbert was looking for was something else, especially with things like his Entscheidungsproblem. — ssu

I have no problem with this. My point was that logic is necessary in all sciences. Of course, the amount of empirical data and the role of hypotheses varies widely. The point of my classical mechanics example was that it is a closed, axiomatic structure, within which one may deduce theorems in the same way that one deduces them in math. Still it is not math, and it is not true in any absolute sense.

What you are making is a hugely reductionist argument that everything has to be deduced from the same axioms. — ssu

Not at all. I believe in open philosophy -- the idea that we should be open to all sources of truth and not restrict our inquiries with a priori assumptions or conceptual spaces. I do, however, see each mathematical theory as defined by its axioms.

If something doesn't fit to be the universal foundation, in your terms it has to be false and whole fields have to be false. — ssu

That is precisely the notion I reject.

I am saying that axioms are no different than any other claims. They are either justifiable, or not. Either adequate to reality (true) or not. Mathematics cannot be exempted from epistemological scrutiny just because it has a canonical, axiomatic form.

There being quantum mechanics or geometries of spheres etc. simply don't refute one another and make the other untrue or false. What is only wrong is the reductionist idea that everything can be deduced from one system or the other. — ssu

Of course the fact that classical mechanics fails to predict phenomena at quantum or relativistic scales, means that it is inadequate to these kinds of realities and so false in an absolute sense.

Geometry is a little different, as it lacks operational definitions of basic concepts such as <straight line>. If we define "straight" to be the path taken by a light ray, then Euclidean geometry is inadequate on a cosmic scale and so false.

I am not a reductionist. For example, biology cannot be reduced to physics because some of the contextualizing data that is abstracted away in physics is the data on which biology is built. So, we must continually return to reality, to the experience of being, to correct our conceits.

Math isn't like this. Mathematics has for example incommensurability, which is totally logical. — ssu

I have no idea what you mean by "totally local." Are you claiming that the concept <incommensurability> came to be independently of any experience of reality? History would seem to rebut this.

It seems to me that you are looking at final canonical forms and forgetting the mental processes that got us there.

Yes, well, crocodiles and dragonflies have some degree of awareness, but zero intellect! — Wayfarer

They have a degree of responsiveness that seems fully explainable neurophysiologically. We have no data implying such animals can actualize intelligibility as opposed to sensibility. I disambiguated my use of "awareness" by saying that it was the same as Aristotle's agent intellect -- whose function is the actualization of intelligibility.

the problem boils down to the fact that consciousness is intrinsically first-person, something of which one is subjectively and immediately aware, or rather, 'that which is aware', and as such is never an object of experience (except for by abstraction). The precise reason why Daniel Dennett refuses to accept that it's real, is because it's not an object of experience. — Wayfarer

Yes, and no. I agree with most of what you say, but we would not know we were aware if we did not experience our own awareness. The problem, then, is not lack of experience, but lack of third person experience. Dennett rejects consciousness because he is unwilling to credit first person experience.

Logical positivists used to say consciousness/awareness was not intersubjectively available. It is. What is unavailable is multiple subjects observing the same token of consciousness. Science is not concerned with token availability, but with type availability. Natural science experiments are intersubjectively available even if only one person at a time observes a result -- as, for example, in the Rutherford scattering experiment. So, the number of observers of result token is irrelevant. What is relevant is type replicability -- and that is intersubjectively available.

Thank you for very lucid explanation. — Wayfarer

You're welcome.

I have the idea that numbers and other intelligible objects are not existent (as are sensory objects), but that they are real. Numbers do not come into or go out of existence, and when we know them, we know them purely intelligibly, i.e. they are only discernible to a rational intellect (which is the thrust of the passage in Augustine). — Wayfarer

Yes, Augustine was inclined to Neoplatonism and that's a powerful intuition. I just don't see evidence to support it. It seems to me that we should rule it out on grounds of parsimony -- we don't need it to explain how we know mathematical truths.

But they're not really objects, they're constituents of thought - so the word 'object of thought' is in some sense a metaphor. (I regard 'objects' as exactly that - things that you have a subject-object relationship with, i.e. everything around you.) — Wayfarer

I agree with most of this, but "constituents of thought" bothers me. While we often reify ideas, it seems to me that the idea <apple> is simply the act of thinking of apples, not thing that can have constituent parts. Still, ideas can have multiple notes of intelligibility that can be separated in thought.

I tried Googling "Augustine intelligible objects" but only found secondary sources so far.

If you justify the axioms, then the justifications will become the new axioms. — alcontali

Only if the justification is axiomatic. It is not. See my new thread on the foundations of math.

As Aristotle wrote: If nothing is assumed, nothing can be concluded. — alcontali

Citation? We need not assume what we know by experience.

I subscribe to mathematical Platonism. However, for practical reasons, I do not make use of the possible link between the real, physical world and the abstract, Platonic world of mathematics. I rather leave this link unspecified. In fact, so does everybody else. — alcontali

Which is why this irrational belief continues to find adherents in mathematics.

First, sciences do not establish their own principles — Dfpolis

Where do you imagine these principles come from? — Fooloso4

Each field of math assumes its principles (its postulates and axioms), but that does not mean that the principles can't be investigated and justified by nonmathematically. I have said how the principles of math derive -- mostly via abstraction, some hypothetically. Investigating and justifying these means is outside of the scope of mathematics and the axiomatic method.

After a full paragraph on Platonism you said:

Platonic relationship 2 + 2 = 4 — Fooloso4

Please read sentences in context.

I said most of the foundations are the result of abstraction. — Dfpolis

To say what they are the result of is not to say what they are — Fooloso4

What they are is not my present interest.

The most basic concepts of of Western mathematics underwent a fundamental change with the origin of algebra, that is when numbers were replaced by symbols. Which leads to the question of whose mathematics? — Fooloso4

The concepts that existed before the addition of unknowns, variables, functions and distributions continue in use today. Adding new concepts does not vitiate old concepts.

Which leads to the question of whose mathematics? — Fooloso4

Mathematics is not personal property. It is an intellible whole that becomes increasingly actualized (actually known) over time. At any time some people know more than others. That does not make them owners, but knowers.

Are you disagreeing with his reading of Aquinas? If so, where do the mistakes lie? Or is it that you are disagreeing with Aquinas? — Fooloso4

As I have thought about the topic, but not read Maurer's paper yet, I am not in a position to say where the errors originate. If they are in Aquinas, they would not be my first disagreement with the Angelic Doctor.

God willed the being, Humans, not God, are the direct efficient causes of sentences by reflecting on the intelligibility of being. God is not the direct efficient cause of the sentences, and so did not will them, but their causes. i explained this in my last reply by saying God was the remote cause by willing both the known object and the knowing subject.

But then God could have willed being so that it's other than it is, right? — Terrapin Station

If God willed "something" other than being, God would will no-thing. That is the whole point of the ontological principle of excluded middle. Being is not the kind of thing that admits of intrinsic change.

"Being" is not a definable term, as it is intrinsically indeterminate. The only way to "change" it would be to make it what it is not, viz. either not-being or something determinate -- so replacing being simpliciter with a determinate kind of being.

Once again, the title of your topic is "The Foundations of Mathematics". Those foundations are not in modern mathematical theory or methodology. Greek mathematics is part of that foundation. Greek mathematics is not "Platonism". — Fooloso4

First, sciences do not establish their own principles, so it would be very surprising if math did. So, we agree on the first part.

Second, I did not claim that Greek math was Platonism. So, I have no idea what you aim to show by the last sentence.

To say:

... our mathematical concepts have a foundation in reality. — Dfpolis

Is like saying a building has a foundation in the ground. It says nothing about that foundation. — Fooloso4

If that were all I said, your criticism would be justified. it is not all I said. I said most of the foundations are the result of abstraction. In response to questions, I went on to explain how that worked. I also said that the rest were hypothetical -- and some of those were falsifiable and the rest not.

Instead, it would be more useful to direct the reader to Maurer's "Thomists and Thomas Aquinas On the Foundation of Mathematics", available free online — Fooloso4

Thank you for the reference, I did not know it.

I disagree with much of the quote you gave from Maurer. Every degree of abstraction is grounded in intelligible reality. It is true that there are no prefect triangles, etc. in empirical realty, but abstraction can leave behind the intelligibility of defects. I don't think anyone has ever come to the idea of a triangle without experiencing an imperfect instance in reality. The same is true of the other examples Maurer gives.

So re those three statements, are they the case because God willed it so, or are they prior to God so that God has no choice in them, either? — Terrapin Station

Neither.

Nothing is prior to God, because if something were, God would be dependent on it, and so not self-explaining, Further, statements are the expression of discursive (time-sequenced) thinking, and as God is unchanging, God does not engage in discursive thinking.

The statements are the result of finite, discursive minds expressing a partial understanding of being. As all being is willed by God, His will makes the statements possible in two ways: (1) by creating intelligible being to be understood and (2) by creating discursive minds to understand it.

In sum, God wills being, and the statements are the result of humans grasping part of the intelligibility of being.

You have completely ignored the foundation of Greek mathematics which makes your pseudo-problem of counting disappear. 2 + 2 = 4 is not a "Platonic relationship", at least not for Plato or the foundation of Greek mathematics. — Fooloso4

And that relates to my OP how?

First of all, with 'scientific' we describe that we are using the scientific method, an empirical way to make objective observations, experiments, tests or measurements, about reality, the physical world as you mention, to solve if our hypothesis are correct or not. Mathematics is logical system. — ssu

That is certainly the modern usage, but not the only one. Traditionally, scientia meant an organized body of knowledge -- organized in terms of explanations reducible to first principles. So, I would say that mathematics is a science in the sense of being an organized body of knowledge -- and that knowledge is an understanding of reality.

Second, "logical system" needs more explanation. All sciences proceed logically. I suspect that you mean a "closed system," i.e. one that simply elaborates an axiom set. Clearly, mathematics does more than that. I agree that mathematicians seek to put their science into a canonical form which is axiomatic, but they also have a history of examining and adding to their axiom sets. For example, we have the questioning of the parallel postulate and the development of non-euclidean geometries, the development of set theory, and discussion of the axiom of choice.

I'd suggest that the finished form of a science is a poor starting point for examining the nature of that science. Isn't it better to reflect on the process leading to the canonical form?

Also, "logical system" is inadequate if there are truths within the scope of the science that cannot be deduced axiomatically. The very existence of such truths implies that there are means of knowing truth more fundamental than the system's logical/axiomatic foundations.

Applicability of mathematics to the physical world isn't the logic that glues mathematics into a rigorous system, but logic itself. — ssu

No, math is not logic. That was Hilbert's view and Godel killed it. Math provides physicists with a set of assuredly true premises, which, combined with other, more empirical, premises will provide a sound conclusion if the empirical premises are true. What is special about mathematical premises is their reliability compared to less reliable Hume-Mill inductions and the hypothesis under consideration. Thus, when an experiment falsifies a conclusion, we can rule out the falsity of the mathematical premises as the cause, and fix our attention, first on our hypothesis, and second on our empirical inductions.

Typically a physicist uses a argument of the form:
Major (mathematical) premise: All systems instantiating mathematical property p are such that they instantiate mathematical property q.
Hypothesis: System A instantiates mathematical property p.
Conclusion: System A instantiates mathematical property q,
where we hope that the conclusion can be experimentally tested. The only "logic" here is a valid syllogistic form (Barbara), not mathematics. Our mathematical knowledge enters as a premise on the same footing as any other premise. For the argument to work, the mathematical premise must be adequate to reality (true).

Above all, something that we have thought to be a mathematical axiom isn't shown to be false from physical reality, but with mathematical logic. — ssu

This is inconsistent with your earlier claim that mathematics as a logical system. In such a system the truth of axioms is unquestioned. An axiom set may prove to be inconsistent, but if it is that only shows that some member of the set is false, not that a specific axiom is false -- unless one has metamathematical reasons for suspecting a particular axiom. But, if one does, then we are justified in examining the truth of axioms, and not merely accepting them as given a priori.

Here you seem to have the idea that if the axiom of choice is independent of ZF, it is somehow 'unscientific' as if other axioms would be the 'scientifically' approved. — ssu

No, not at all. If you go back to the OP, you will see that I divide axioms into three groups: (1) justified by abstraction, (2) falsifiable, and (3) unjustified by abstraction and unfalsifiable, and so unscientific. The question is not whether we are doing math, but how we justify truths in general. It is special pleading to say that all scientific (in the older sense) truths need be justified, but mathematical premises get a pass.

Perhaps the problem here is that some mathematicians see themselves as enlightened by mystic insight into the realm of Platonic truth. I do not.

So, the problem is not that C is independent of ZF, it is that C is unjustified and seems to be unjustifiable. Perhaps I'm wrong on C being unfalsifiable. Perhaps some consequent of C can be falsified. Whether my example holds, however, is a contingent matter,and irrelevant to the need for justification.

To say in this case that all of the math in all of those various fields of mathematics are unscientific is, should I say, out of whack. — ssu

I do not see this. After Laplace published his Celestial Mechanics and dispensed with Newton's ill-conceived "hypothesis of God," physics was seen as axiomatic and given canonical form by mathematical minds such as Hamilton and Lagrange. That did not make it true in any absolute sense. The classical mechanics they developed is still useful and taught today, but it is not true. Unjustified foundations necessarily give unjustified conclusions.

To make my argument short, scientific/unscientific is a poor definition in math, far better would be to speak of logical and illogical. We have had and can indeed still have illogical presumptions (or axioms) of the nature of math, just like some Greeks thought that all numbers had to be rational and were truly disappointed when finding out that there indeed were irrational numbers. — ssu

First, I have problems with the theory of types as a solution to Russell's paradox, but that is for another day.

What validly follows from axioms is necessarily "logical," but if the axioms are unjustified it has little claim on being true. If we understand science as organized knowledge, and what is unjustified does not count as known, then any consequent of an unjustified axiom is unscientific.

What I'm questioning is the notion that an account can be given of intelligible objects (such as number) in purely mentalistic terms. I think that Platonic realism posits that numbers are real for anyone who can count. So they are only knowable to a mind, but they are not the product of an individual's mind. — Wayfarer

Providing a purely mentalistic account is exactly what I am not doing. I am saying that our mathematical concepts have a foundation in reality. If there were no countable beings and no measurable beings, we could have no experience from which to abstract mathematical concepts.

While we may combine concepts in ways not found in reality, ultimately our concepts are traceable to the actualization of intelligibility found in nature. Objects act on our neural system is our senses, presenting intelligibility to awareness, Still, until we turn our awareness to these encoded contents, they are not actual concepts. It is our being aware of contents that makes what was merely intelligible actually known.

Further, engendered concepts are dynamically united to a corresponding intelligibility in nature, to the engendering object. First, the object's modification of my neural state is identically my neural representation of the object (encoding contents I can become aware of). Second, a single act of awareness simultaneously actualizes both the object's intelligibility (making it actually known) and the subject's capacity to learn (making it actually informed). So, there is no isolated mental construct here, but a (partial) ontological penetration of the intelligible object into the knowing subject.

This analysis does not, therefore, make numbers a mental product. It makes them the actualization of a prior intelligibility in nature -- eliminating the need for numbers to actually exist prior to being known -- in other words, the need for a Platonic realm.

think it's reason that naturalism has no account of. — Wayfarer

To me, "awareness" means agent intellect, without which there is no reason or conscious experience.

The link to Augustine is not available. I would like to see it if you have it (message it?). I am a great admirer of Augustine's insight, and think his account of the soul coming to know intelligibility is largely compatible with my understanding of Aristotle and Aquinas.

The page in Britannica is good starting point to answer your objections:

Axiomatic method, in logic, a procedure by which an entire system (e.g., a science) is generated in accordance with specified rules by logical deduction from certain basic propositions (axioms or postulates), which in turn are constructed from a few terms taken as primitive. These terms and axioms may either be arbitrarily defined and constructed or else be conceived according to a model in which some intuitive warrant for their truth is felt to exist. — alcontali

Defining a method is not an argument justifying the application of the method. Clearly, many mathematicians are concerned the justifying their axioms. I am also concerned about this issue. You seem not to be. So, we do not share a common interest.

You continue to make unargued claims. You are attacking Platonism, which I do not hold. You are attacking my claims, not by offering a substantive critique, but by the ad hominem that I am unqualified to comment, despite having studied advanced math and its history. So there is no point in our continuing to dialogue on this topic.

Yes. (1) Whatever is, is, and whatever is not, is not. (2) Something must either be or not be. And, (3) nothing can be and not be in one and the same way at one and the same time. These are reflected in the kinds of assertions that can be true.

Still, these are not constraints, because they exclude no-thing.

Yet, Godel's work shows more: it shows that there are truths that cannot be deduced from any knowable set of axioms. — Dfpolis

Does the Incompleteness Theorems say really this? Correct me if I'm wrong, but doesn't (the first Gödel Incompleteness Theorem) say that for any 'set of axioms' or consistent formal system there exists specific true but unprovable statements. That's a bit different — ssu

It does not seem different enough to vitiate my point. In any lifetime, or finite number of lifetimes, we can only go through a finite number of axiom sets. So, there are true axioms we cannot deduce. Or, am I missing your point?

I think maybe "unscientific" in this context is wrong usage. In any case, mathematics is not an experimental science. — tim wood

I agree here with tim wood, talking about scientific/unscientific here with foundations mathematics is totally out of whack. — ssu

Is it? If we cannot justify certain axioms, how can we rely on the conclusions? As a physicist, I want my mathematics to be not merely consistent, but applicable to the physical world. We know that, however reasonable, the parallel postulate is not so applicable if we define straight lines as geodesics. How are we to know that the consequences of ZFC fair any better?

So, you will have to explain why my criticism is "totally out of whack."

What is the axiom of choice? — Noah Te Stroete

The Wikipedia provides a good discussion. It says "Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if the collection is infinite."

The basic problem with it is the same as that with the parallel postulate -- we have no experience of the infinite per se, from which to abstract it.

Personally, it seems reasonable, but then so did the parallel postulate when I studied geometry. My criticism is based on purely its lack of epistemological justification.

Thank you for your comments. I have no problem with the neoplatonic One Identified as God.

So do you think the law of the excluded middle, or the Pythagorean theorem, only came into existence with h. sapiens; or that such principles are eternal, and are discovered by any intelligence sufficiently rational to discern them? — Wayfarer

I am not positing that we're the only rational animals. I am saying that the principle of excluded middle reflects the nature of being and so is intelligible to anything capable of abstracting being as being. I am not saying that the Pythagorean theorem is true in general, only in flat spaces (where the parallel postulate holds).

I think the error you're making with the 'Platonic world' is to try and conceive of it as a literal domain. But what of the 'domain of natural numbers'? — Wayfarer

If someone wishes to define a "Platonic world" in non-literal way, I will be happy to comment on their effort. If it's undefined, it is irrational to appeal to it as an explanation.

The domain of rational numbers is countable objects. It is unlimited because we can partition unities into countable parts indefinitely.

Surely that is something real, as real numbers are included in it, and irrational numbers are not. — Wayfarer

Irrational numbers are not based on countable objects. That does not mean that they are not based on other aspects of nature, i.e. measurable quantity. In neither case do actual numbers exist in nature. What exists in nature is the potential to be counted or measured. Actual numbers exist only in minds actually thinking them.

And how does your account differ from run-of-the-mill evolutionary naturalism, in which there is nothing corresponding to what Aquinas deem the soul, which is 'capable of existence apart from the body at death'? Your account most resembles that of John Stuart Mill, whom I'm sure would not be the least inclined to agree with Aquinas. — Wayfarer

I distinguished abstraction from the Hume-Mill model of induction in my previous response (to Tim Wood). I wouldn't mind if my account were based on the aspects of reality fixed upon by naturalists, but, in point of fact, it does not. It requires the operation of an intellect in act (Aristotle's agent intellect) to make what is merely intelligible actually known.

Naturalism has no problem describing neurophysically encoded contents, but it has no rational account of awareness. Daniel Dennett showed in Consciousness Explained that naturalism cannot account for the experience of consciousness, and David Chalmers has pointed out the difference between the kind of progress made by neuroscience and the "hard problem of consciousness." In a previous thread I discussed the difference between intentional and material reality.

So, any account that hinges on the actualization of notes of intelligibility by awareness (as mind does) is beyond the scope of evolutionary naturalism.

Do you know that Godel considered himself a mathematical Platonist? — Wayfarer

No, I did not, but as many mathematicians are, it's not surprising. Being a good mathematician doesn't make one a good philosopher.

and surely we cannot abstract universal truths that are not instantiated. — Dfpolis

Are you sure of that? Are not all instantiated truths particular truths, and are not all universal truths abstract? You have left out how to get from particular to "universal." And how do you instantiate the truths of transfinite arithmetic? — tim wood

Yes, I am sure.

Yes, to be instantiated is to be particular. We move from the particular to the universal by removing particularizing notes of comprehension. For example, we ignore that we are counting apples, that apples are fruit, etc. and fix on the noted that matter to counting, i.e. that we are dealing with an instance of a unit and that unities are countable.

Contrast this with the Hume-Mill model of induction. In it you see 100 black crows, no white crows and posit "All crows are black." Whereas abstraction is a subtractive process, Hume-Mill induction is an additive process. We add the assumption that the cases we have not seen are like the cases we have seen. In abstraction, we add nothing. We merely remove notes of comprehension that don't interest us.

Re transfinite numbers: We come to the notion of Aleph-0 (countable) infinity by noting that the counting process has no intrinsic limit. We come to the notion of Aleph-1 (uncountable) infinity by proving that the numbers we assign to the points of continuous extents cannot be counted. This was done by Georg Cantor in his 1874 uncountability proof.

Further, it may be that we can trace to a ground in nature, but is that all? — tim wood

I am not saying that nature is all that is, only that we have no reason to posit a Platonic realm of ideas.

And while the idea of concepts needing minds to have and hold them informally and intuitively seems right, is it altogether right? — tim wood

It is, unless you have a different definition of "concept," but then, you're not talking about what I'm talking about.

the content of thinking - the that that is thought about, is pretty much always already there prior to a mind thinking it. — tim wood

Yes, the content may well be "there" as intelligible, as something capable of being known, but not yet known -- and so as not yet a concept. That's what it is for a concept to be grounded in reality.

This means there is an accounting/definition problem. — tim wood

Specifically?

I think maybe "unscientific" in this context is wrong usage. In any case, mathematics is not an experimental science. — tim wood

Well, that is not how the game is played now, but that doesn't mean the game is played rationally, does it? The value of Popper's falsifiability criterion is that it restricts hypotheses to ones we can gain intellectual traction on. If you allow hypotheses that cannot be tested, then any guess, no matter how irrational, can be posited.

Of course, the test might not be experimental. While Godel's work means that we cannot prove the consistency of a set of axioms, it doesn't prevent us from proving their inconsistency. So, we could have mathematicians deducing hypothetical consequences in the hope of finding an inconsistency. If they do, they'd have proven an axiom was false. But, if they don't, they won't know that any of their hard-won conclusions are true.

As you note, we can't prove that the axiom of choice is true or false in the context of ZF. That leaves it unfalsifiable.

As for "games," what would you call playing by rules that are either false, or ungrounded in reality?

the pursuit of the consequences of which, which may be "no different in principle," are in practice and in fact altogether and entirely different from a mere game. — tim wood

What would the difference be? It's a game many mathematicians enjoy?

Being is some way(s) rather than other ways, no? — Terrapin Station

I'm not sure what you mean. Surely anything that can act in any way must be, and nothing can be unless it can act in some way -- for it it could not, it would be indistinguishable from no-thing.

What determines the nature of being--God, or.something else? — Terrapin Station

i think the question involves a category error. Being is in indeterminate, not determinate. As what essences limit and so determine, existence is not itself determinate. It is the indeterminate power to act.

Yes, I did miss it. It did not appear in my email notifications. Thank you for your thoughtful reply.

Let me say that, while I am Cristian, I don't think that we can have proof that God acts in a Christian way - that we have no proof of a "Christian God," or a Jewish or Moselim God for that matter. Belief in such a God is a matter of faith. It may be justified internally by some more or less intimate relation. Knowing that relation (an I-Thou) points to a Thou, making faith a genuine, but incommunicable, kind of knowledge. It may be justified externally, not by empirical knowledge, but by the worthiness of the commitment it calls for.

Still, I think the logic of arguments such as those of Aristotle, ibn Sina and Aquinas (and the one I posted here) is sound. Just because what logic tells us is limited, and perhaps even inadequate to a well=lived life, does not mean that it is false. So, I think we must agree to disagree. I think that we can prove that there exists a timeless Power maintaining the universe in being.

The amount of mathematics used by physics does not change its fundamental nature. It certainly does not turn physics into mathematics. It just makes sure that it is incredibly consistent. It is its consistency that explains its success. — alcontali

You are missing the point. Deductions are only sound if the premises are true and the logic valid. According to you, no mathematical proposition is true. So any argument a scientist makes using a mathematical premise is necessarily unsound.

The difference between physics and mathematics is not that one is about nature and the other not — Dfpolis

That is exactly the difference. — alcontali

If you are only going to repeat you faith claims, and not try to justify them there is no point in posting on a philosophy forum.

Math is about nature as quantifiable — Dfpolis

Mathematics is not number theory. Most mathematical theorems are not about numbers or quantities. — alcontali

I did not say it was number theory. This is not the first time you have twisted my claim, saying I was talking about number theory when I did not mention it. This is not honorable. I named other areas of math, and explained how set theory was based on nature as quantifiable.

You can represent a set by its membership functions and disregard what elements it contains. From there on, the paradox becomes a problem with these membership functions. — alcontali

Again, doing so leads to a contradiction precisely because proceeding in this way makes assumptions that are not based on abstraction.

Since you did not respond the main argument of my last post, there is no point in continuing. I am simply wasting my time.

I skipped a number of points as they did not address my argument in a substantive way, and so need no response.

Third, in my proof infinite being does not stand as unexplained, but as self-explaining and precisely because it is infinite being, so that what it is entails that it is. — Dfpolis

This is an equivocation. Either you can explain the existence of God, that is provide a discursive explanation or you cannot. You have not. — Fooloso4

This is false. I explained, discursively, why God's essence (what He is) entails that He is.

You claim that there is an:

Infinite being [who] can act in all possible ways in all pos­sible places at all possible times. — Dfpolis

and build your discursive explanation based on that assertion. — Fooloso4

This is false.

There is no point in spending more time responding to you when you do not understand and respond to what I actually write. Two false claims are enough.

So, the order of precedence here is God -> created being (including humans) -> logic (created by humans). — Dfpolis

So if logic is simply something created by humans to think about reality, then God would not in any way be constrained by logical possibility, right? — Terrapin Station

Logic is as it is because, to be salve veritate (truth preserving), it has to reflect the nature of being. Being is not a constraint, because being only excludes non-being -- which is to say being excludes nothing. What excludes nothing is not a constraint.

God, then, is limited to the possible, the which He cannot instantiate himself - like eating a sandwich - so he acts through agents - demi-urges? Demons? Lesser deities? is there a problem with the divine/common interface here? — tim wood

I eat sandwiches for God, and so do all the other sandwich eaters. We can only eat because we exist, and we only exist because we are divine activities.

What is "contradictory' cannot be the same as the possible and not-possible, beacuse the latter is mutable, changes over time. — tim wood

As I discussed in my thread on the analogy of necessity, there are different kinds necessity and possibility depending on the contextualizing basis. What is physically possible can change over time, as the initial conditions do. It could also be different in other universes with different laws. What is ontologically possible is not like that. Ontological possibility is only limited by the ontological principle of contradiction. If a fully specified A is, then it cannot not-be That never changes.

There are some very appealing and intuitively obvious answers, but those cannot be our criteria - if for no other reason than the question relates to the capabilities of "infinite" beings — tim wood

That is why we need to be very careful with out definitions.

In any case, we've devolved this notion of "God" from an omnipotent and infinite being to one who cannot do anything! — tim wood

False. God can do anything. The problem is that contractions cannot be things

I like ham, but can you do pastrami? — tim wood

I'm an omnivore..

It strikes me that the only possible act that God engages in directly is the act of creation ex nihilo. — Theorem

Creatio ex nihillo was in the past. One an ongoing basis God engages in creatio continuo -- mainting conteinfent beings in existence.

it would imply that God's existence and the existence of some logically possible universe are mutually dependent. In other words, if God exists only when he is exercising some capacity, and if the only capacity he has is for creation ex nihilo, then God exists iff some logically possible universe of his own creation exists. — Theorem

God primarily maintains His own being. Thus, Aristotle called Him, "Self-Thinking Thought." This is an immanent activity -- directed to self-perfection as opposed to a transient activity, which is directed to others. So, God's immanent activity is contingent on nothing else.

The necessity of God upon the universe is not ontological, but epistemological. Knowing the universe is sufficient to lead us to know that God exists by necessity.

No, we're not going to get stuck. I'm not saying that your concept of a brick is whole brick. It is a projection of the brick. (Think of a projection of power.) The fact that the brick is acting on and in you by modifying your neural state does not mean that the whole brick is in you. Only a subset of what it can do is in you. Still, it is acting in you.

We think of physical objects as having well-defined boundaries, but that way of thinking does not exhaust their reality, They are surrounded by a radiance of action: they have a gravitational field, scatter light, and so on. This radiance of action is as much a part of their being as their core. If we took it away, they would no longer be the same. They would be something different. It is this radiance of action that penetrates us in modifying our neural state.

I do not intend to provide detailed replies to each of your posts, which have become repetitive. Instead, I will simply read them to see if you've answered any of my points.

'll begin by summarizing my position. I agree that mathematics does not seek to justify its axioms. This point was made by Aristotle 2500 years ago. That does not mean that most of its axioms are not justified. They are just not justified by mathematics.

We may divide the axioms into three classes.
1. Most axioms are abstracted from our experience of nature as countable and measurable. You have agreed that this is so historically and have offered no reason why is not true today. To be concrete, children learn to count by counting particular kinds of things, but soon learn that the act of counting does not depend on the kind of thing we are counting, only that it be countable. Thus, they abstract concepts such as unit and successor from the experience of counting real-world objects. This is the empirical basis of arithmetic and its axioms.
Since we are dealing with axioms abstracted from, not hypothesized about, reality, there is no need for empirical testing for them to be known experientially. Further, since the axioms are instantiated in reality, which cannot instantiate contradictions, we know that such axioms are self-consistent without having to deduce their self-consistency.
As we can trace our concepts to experiences of nature, and since there is no evidence concepts exist outside the minds of rational animals, there is no reason to posit a Platonic world. Doing so is unparsimonious and irrational.
2. Some axioms are hypothetical.
a. Some hypothetical axioms can be tested, e.g. the parallel postulate. You have not objected to my claim that the parallel postulate can be tested by measuring the interior angles of triangles.
b. The remaining hypothetical axioms can't be tested, e.g. the axiom of choice. These are unfalsifiable and unscientific, We agreed that unfalsifiable hypotheses are unscientific. I have pointed out that as, unscientific, pursuing their consequences is merely a game, no different in principle than any other game with well-defined rules, such as Dungeons and Dragons.

Against this you claim that " Axioms can best be considered to be arbitrarily chosen." Best on what basis? What is optimizedt? If the axioms are formalizable in arithmetic, we have no way of knowing that they are even self-consistent. On my account we do. Surely it is better to know we are dealing with a self-consistent system than to waste a lifetime on what may turn out to be utter nonsense. So, how is your notion "best"?

As any knowledge abstracted from nature can be applied to nature, there is no problem with physicists using true mathematics to deduce conclusions about nature. Physics do this routinely. On you account this would be a grave error, for it would be mixing premises of indeterminate truth with premises that are true empirically. Yet, mathematical physics is one of the most successful sciences. Your theory can't explain this success. On it, what mathematical physicists do is completely unjustifiable.

The difference between physics and mathematics is not that one is about nature and the other not, but that they are about different aspects of nature. Math is about nature as quantifiable (countable and measurable), while physics is about nature as changeable, and mathematical physics is about the quantitative aspects of nature as changeable.

You object that we can prove nothing of the real world, but have provided no rational for this. You have not explained why I can't prove there are two pieces of fruit in a bowl, or that two objects and two more objects are four objects. Even if there were the mythic "Platonic World," that wouldn't mean abstract concepts can't be instantiated. In fact, the only way we learn concepts is by abstracting them from their instances. So, no more dogma. Let's have a proof if you have one.

You object that set theory is not about nature as countable, but it is. It's just not about it in the same way as number theory. This is because sets are collections of objects, and, in the context of set theory, "object," "unit," and "element" are convertible terms. That is why sets have cardinality.

The reason for Russell's paradox is not some formal problem that requires a theory of types (though a theory of types avoids the problem). The reason for it is that there is nothing in reality from which we can abstract the concept of the set of all sets that do not include themselves, just as there is nothing in reality from which we can abstract the parallel postulate or the axiom of choice. In other words, there is no (actual or potential) well-defined collection of objects (note the real-world reference) that is the set of all sets that do not include themselves.

I know that you will object that the "objects" in the definition of "set" need not be physical, but I did not claim that they were. They can be intentional beings, i.e. concepts in the minds of real persons.

Clearly, we may not believe (accept) what we know, which would be impossible if knowledge were a species of belief. — Dfpolis

If you know it, it means that you can justify it. So, why would you not believe it? — alcontali

Belief need not be rational. People know they can't afford something because they know their financial situation, but buy it anyway because they want it. They allow their desire to convince them that they can afford it. There are thousands of examples of desire-based beliefs overriding known facts. Knowledge cannot be a species of belief because we can know one thing and believe the contrary. Plato did not want to acknowledge this, but it's true.

If we only need begin with unjustified axioms, we can start with any assumptions and prove anything. — Dfpolis

No. A system becomes trivialist because it contains a contradiction, for example — alcontali

But, confining ourselves to the formal approach you champion, we can't know that any system formalizable in arithmetic is self-consistent. So, almost any system may be trivial on you account. You need to do better. All you're doing is ruling out obvious nonsense, leaving open the possibility that all mathematics may be obscure nonsense,

Math does not justify axioms by experimental testing. In fact, Math does not justify axioms at all. If you justify axioms by experimental testing, then it is simply not math. In that case, you are doing something else. — alcontali

We agree, You're doing something more fundamental than a science when you examine the foundations of a science -- metamathematics or metaphysics, for example.

I personally do not believe that a good physicist could ever be a good mathematician, nor the other way around. — alcontali

Poor Pierre-Simon Laplace! Poor Carl Friedrich Gauß! Poor Jules Henri Poincaré! Poor Emmy Noether. Poor John von Neumann! If they'd only had your insight, they might not have wasted their time doing both. As with your statement about how Euclid influenced Socrates, you seem to like beliefs that ignore history.

Concerning the coherence theory of truth, I agree with Bertrand Russell's objections: — alcontali

I said nothing about the coherence theory, which I reject.

Therefore, I cannot agree with "Newtonian physics is true with respect to" — alcontali

Non-sequitur -- as it is based on attacking a position that is not mine.

Entanglement allows for simultaneous being and not being in the real world. — alcontali

You have no idea of what "entanglement" means, do you?

I am tired of this. Do some reflecting on what I said.

Goes to show how barren theology has become, when modern arguments for God are nothing more than restated millennium-old syllogisms — Maw

If a proof is sound, there is no shame in restating it. Do you have a substantive criticism, or is your only objection that my argument is not in vogue?

That is why I provided a proof. — Dfpolis

Call it what you like but it is nothing more than a claim for the existence of a being whose existence you assert but cannot prove or demonstrate exists. — Fooloso4

When you have rebutted my argument, you may claim this. Without pointing out a false premise or a logical misstep, this remains your unsupported belief.

Do you have a citation for Aristotle? — Dfpolis

No. — Fooloso4

Then you should not claim the authority of Aristotle.

Claiming that a being is the cause of being leaves unexplained the existence of that being. — Fooloso4

This mischaracterizes the argument. First, Aquinas points out that "being" is not predicated univocally of God and empirical beings. Instead, God is called a being because God is the source of empirical being. This is an analogy of attribution. The example is that food is healthy, not because it is in good health, but because it is a cause of health those who consume it. So, the source of empirical being cane be called a being, not that use of "being" is not the same as the "being" in empirical being.

Second, it is not being in the abstract that explains being in the abstract, but a particular, infinite being that explains other finite beings.

Third, in my proof infinite being does not stand as unexplained, but as self-explaining and precisely because it is infinite being, so that what it is entails that it is.

Claiming that there is self-explaining being is not to provide a discursive explanation. — Fooloso4

I made it clear in the OP that I was not talking about discursive explanations, but about dynamical ones. Still the fact that God's essence is His existence is the discursive reason He is self-explaining in the dynamical sense.

You simply posit what you cannot explain or demonstrate. It is just kicking the can. — Fooloso4

Then you will have no trouble pointing to a false premise or an invalid logical step.

Aquinas wrote for a more philosophically literate audience -- one that knew the distinction between essential and accidental causality. — Dfpolis

You should not underestimate your own audience. There may be some here who do not know the difference but some who do. — Fooloso4

It is still better to avoid confusion by a judicious choice of terms.

Positing a necessary being or, facts as you would have it, explains nothing. It is a misuse of the term explanation. I think you might know this and that is why you called you assertion a fact. — Fooloso4

I did not posit, I offered a proof. A proper critique would point to specific errors in what I offered.

While there are some who still attempt to defend Aquinas' argument others, including theologians, have rightly moved on. Your argument fares no better than his. — Fooloso4

Then what, specifically, are my errors?

I have spent a lot of time responding, but very little of what I have responded to is critical of my actual argument. The fact that many are confused about Aquinas's arguments is not a criticism of what i said.

As for your post on Aristotle, I will not respond to it, as it would take too much time.

The issue is that your distinction between infinite and finite beings is made in terms of an ambiguous definition of "possible acts". — Theorem

Not quite. Infinite being can effect any possible act either directly, or by indirection. While God can't eat a ham sandwich Himself (as that would entail finiteness), God can create a being who can. So, God can effect any possible act, while a finite being cannot. In other words, there is no barrier to effecting the act, there is only a problem if one over-constrains the act so as to make it instantiate a contradiction.

Using this line of reasoning, we could say that a finite being acting as only an infinite being or as only any other finite being can is also not a possible act. Therefore, finite beings can engage in any possible act. — Theorem

I hope that I have resolved this to your satisfaction above. God can do anything that does not instantiate a contradiction, because it is not possible to instantiate a contradiction. Finite beings have intrinsic limits to their power to act, so that there are possible acts not within their power.

Agreed? — tim wood

Yes, but the idea of a particular brick requires that that brick modify your nervous system to create a representation you can be aware of. So, by acting in you, the extramental brick penetrates you (your neural representation is identically the brick modifying your neural state). It is this shared being that makes knowledge possible.

Your awareness of the brick's activity is your idea of the brick. Aristotle points out that the one act of awareness actualizes two distinct potentials: You capacity to know and the capacity of the brick to be known (its intelligibility). So, again knowledge is based on subject-object unity -- this time joint actualization.

What you may regard to be the relationship between thought and reality is simply your thoughts on that relationship. A clear example of why your simplistic bivalent logic fails: — Fooloso4

No, not just my thoughts, but those of a community of scholars who have been investigating the issue for 2500 years. But, even if they were mine alone, noting that does not rebut my analysis. To do that you have to deal with what I said in a substantive way,

... the opposite of red is not-red ... — Dfpolis

What is the opposite of red? Is blue the opposite of red? Is green or yellow? — Fooloso4

I said what the opposite is in the line you quoted: not red -- not this or that kind of not red, but anything not red, aka the privation of red.

If God exists (something like the typical ideas of God re the Judeo-Christian God), then either:

(a) God created logic, or it's at least part of His nature, and God could make logic however He'd want to make it--He has control over His own nature,

or

(b) Logic is more fundamental than God, and God can't buck it any more than we can. God must conform to it. It supersedes Him in its regard. — Terrapin Station

Logic is a science which provides rules for mediated thought about reality. God does not have mediated thought, but knows all reality immediately in his act of sustaining it being. Therefore God does not need or use logic.

So, I simply deny both horns of your supposed disjunction. God is unqualified being. The beings of experience depend on God. Humans develop logic to think about being in a rational way. So, the order of precedence here is God -> created being (including humans) -> logic (created by humans).

Continuing...

In science, the observations are the P (justifying statement) and the theory (knowledge statement) is the Q, in P => Q — alcontali

This shows a disqualifying lack of understanding of the scientific method. There is no case in physics, or in any other science, in which observations logically imply a theory. Observations are particulars, while theories make universal claims. The implication you suppose has an undistributed middle and is necessarily invalid,. This has been clearly understood since at least the 1230s, when Robert Grosseteste wrote canons for the scientific method as it exists today.

P does not affect the arrow, which is the real knowledge. — alcontali

While I agree that conditionals can express truths independently of the truth of their antecedents, they need not be the "real knowledge," whatever that may me, What most scientists seek to know is how their observation sets can be understood in terms of more fundamental universal principles, aka theories, the P in your proposition.

It may be true, for example, that if there were no observers there would be no observations, but few scientists would consider this to be getting to the meat of the matter, to be "the real knowledge,"

Mathematics is not justified by experimental testing, and is therefore, not scientific — alcontali

Yes, and no. The portion of mathematics following from propositions abstracted from nature, or testable by observation (e.g. the parallel postulate), is scientific. The portion deriving from unfalsifiable hypotheses (e.g. the axiom of choice) is clearly not scientific, for it violates the accepted canons.

In his lecture, Gödel and the End of Physics, Hawking spent quite a bit of effort justifying his views. For me, it works. — alcontali

As this is entirely irrelevant to the OP, I shall leave you to it.

While physics can be and has been axiomatized (e.g. quantum theory and quantum field theory) — Dfpolis

If it is physics, it is about the real, physical world, and in that case, you can test it. Therefore, it will not be accepted, as a matter of principle, that it does not get tested. — alcontali

You are missing the point. I am not saying that we shouldn't test the axiomatic foundations of physical theories, but that our capacity to investigate those foundations shows that being an axiom does not preclude justification. Your response shows that you agree, but want, for no stated reason, to exclude mathematics from the fields whose axioms can be investigated and potentially justified. That the parallel postulate was suspected from the beginning, and the uncertain status of the axiom of choice shows that your views are hardly universal.

So, a bowl that holds only one apple and one pear cannot be proven to hold two pieces of fruit? — Dfpolis

No. It will undoubtedly be true, but it will not be provable. — alcontali

This alone is sufficient to reject your views. We can prove it by (1) noting that apples and pears are both fruit, (2) that they are also both units, and (3) applying ordinary arithmetic via the dictum de omni. Feel free to rebut this.

So, 2 objects and 2 more objects might not yield a total count of 4 objects outside the visible universe? — Dfpolis

Doesn't matter, because you cannot observe it. Therefore, without observations in an experimental testing fashion, such claim about the non-visible universe is unscientific. — alcontali

This is irrational and inconsistent. You claim that mathematics need not be justified by observation. I hope you would agree that it is a mathematical truth that 2 + 2 = 4. By the dictum de omni, this is true if and only if it is true in all instances. Whether the instances are observable or not is irrelevant.

Further, I do not accept your restriction of science to fields that employ the hypothetico-deductive method. You cannot define you way to a conclusion about reality. If 2 + 2 = 4, it does so always and everywhere, not merely in some domain you arbitrarily choose to define.

Mathematics requires you to painstakingly construct the world in which you will derive your mathematical theorems. We did not construct the real, physical world. Therefore, we are not allowed to derive mathematical theorems in it. — alcontali

Mathematicians construct no worlds. They merely work out the implications of axioms that may or may not be justified by our experience of the real world. If the axioms are justified, those implications will be applicable to the world from which the axioms are derived. If the axioms are unfalsifiable hypotheses, those applying them are merely playing complex mental games. They are entitled to play their favorite games, but they can hardly expect society to support their play.

I only wanted to refer to the fact that scientific theories are enumerable. — alcontali

Sentences are enumerable, but I don't think theories are as they may contain unspecified constants that are indenumerable. Also, it is unclear that the judgements sentences express are enumerable, as concepts can be analogously predicated. So there is not a one-to-one mapping of concepts to words.

That is probably true for "a science" but not for "science", which is simply any proposition that can be justified by experimental testing. — alcontali

I see no need to restrict systematic knowledge to what can be justified by the hypothetico-deductive method. What is so justified is not known to be true, only known to be justified. Since we do not know it to be true in any absolute sense, it does not even meet the JtB definition of "knowledge."

Yes, agreed. I do not think that knowledge is necessarily a "true" belief, with the term "true" as in the correspondence theory of truth. Knowledge as a "justified belief" should be sufficient. — alcontali

I am willing to agree with this in the context of experimental/observational science; however, we can do better wrt to math, being and certain mental topics.

My preferred approach is to use an analogous definition of truth as adequacy to the needs of a particular discourse. Then, for example, Newtonian physics is true with respect to many engineering needs.

Experimental testing always occurs in the real, physical world, of which we do not have the axioms. — alcontali

While it is quite true that we do not have an exhaustive knowledge of reality, that is not the same as having no knowledge of absolute real world truths. We can and do have axioms applicable to the real world. Recall that the root meaning of "geometry" is "land measure" and many of its axioms are true of real-world geometric relations relations. Number theory derives from counting real world objects, and applies to such operations. We also know some of the principles of real-world existence. No real thing can be and not be in one and the same way at one and the same time, and so on.

Therefore, we cannot axiomatically derive that what can be experimentally tested. — alcontali

Of course we can. We can measure the interior angles of plane triangles and see if the results agree with the prediction that they will sum to two right angles. Then, the result is both axiomatically derived and experimentally confirmed.

Also, as I have mentioned earlier, physicists have axiomatized quantum mechanics and quantum field theory. Special relativity rests on two axioms. So, in al of these fields, we have the ability to proceed axiomatically, but that in no way prevents us from testing our deductions experimentally.

Math justifies by axiomatic derivation, while science is does that by experimental testing. — alcontali

I agree that this is true with respect to justification, if are restricting yourself to sciences that use the hypothetico-deductive method.

If a proposition is derived axiomatically from a set of axioms that construct an abstract, Platonic world, you cannot experimentally test it, because that would require the objects to be part of the real world and not the Platonic world in which they have been constructed. — alcontali

I hate to break it to you, but there is no Platonic world. There is the real world and there are mental constructs that exist in the minds of people living in the real world.

Historically, most axioms have been abstracted from our experience of reality. For example, the Dedekind–Peano axioms are all derived from our experiences of counting and dealing with equal quantities. A very few (principally Euclid's parallel postulate) were not derived from experience. The fact that neither the parallel postulate, nor any equivalent to it (such as the sum of the interior angles of a triangle) could be abstracted from experience has been recognized as a problem from the very beginning of axiomatic geometry. So, from the beginning, we've had axioms that could be abstracted from reality and hypothetical axioms, such as the axiom of choice.

We use the same logic in deducing both predictions from physical hypotheses and mathematical theorems. In fact, many of the axioms used in mathematical physics are identical to axioms used in mathematics. So, the only methodological difference is that physics has a much greater percentage of hypothetical axioms and so makes greater use of experimental confirmation. (While it is not part of the canonical procedure, many geometry students have measured the interior angles of triangles.)

When we test conclusions, we do not test them in abstract, universal form, but as they are instantiated in real-world particulars. So, the fact that we have no experimental access to abstract forms is irrelevant.

The axiomatic method is defined and discussed in numerous places, such as here and here. — alcontali

I am not denying that. I meant that there is no clear distinction between the methods you mentioned. The axiomatic method is no different than the method used in rigorous papers in physics. One states one's premises/axioms and then deduces consequences. The main difference is that in mathematics, hypothetical axioms need not be falsifiable. Those of us trained in the natural sciences do not see unfalsifiable as an advantage, especially given that Godel work ruling out consistency proofs in systems representable in arithmetic.

So, the only way to insure consistency is to abstract one's axioms from reality, which cannot instantiate a contradiction.

After Euclid's Elements introduced the axiomatic method, Socrates got the idea that philosophy had to be approached in a similar manner. — alcontali

Socrates died in 399 BC, Euclid flourished about 100 years later, c. 300 BC. If you read the history of Greek science, you'll learn that Euclid modeled his method on the logical approach developed by Aristotle.

it was not a good idea for science, as would later become clear from Aristotle's now outdated scientific publications, but it works for mathematics and morality. — alcontali

While science moves on, it is hardly a failure to found a number of fields, including political science, logic, mathematical physics and marine biology. Aristotle was a tireless researcher, reading all the works of his predecessors. He was also a thorough observer and empiricist, insisting that his students dirty their hands with dissections and keeping informed on advances in mathematical astronomy. He knew more about viscous fluids than Newton and his work on Aegean fish is still a valuable reference.

Aristotle's approach to empirical science was empirical, not axiomatic. His approach to philosophy was fundamental and logically, but not axiomatic in the sense of positing unexamined assumptions. Instead, he saw the role of metaphysics to be the examination and justification (but not deduction) of first principles.

As a passing note, the axiomatic method does not work for morality, nor did Aristotle claim that it did.

Axioms can be abstracted from reality — Dfpolis

That is how axioms were originally understood: — alcontali

I am glad to find some agreement. Those not so abstracted are, then, hypothetical. If they cannot be tested, they are unfalsifiable hypotheses and highly suspect.

How does the so-called "axiomatic method" justify its axioms? — Dfpolis

It doesn't. In fact, that is even forbidden, because in that case, they are not axioms. — alcontali

This is utter nonsense. Axioms are axiomatic wrt a particular field -- meaning that they are assumed, but not justified within the context of that field. That does not preclude them being justified by a more fundamental field. As we have just agreed, the original justification of mathematical axioms was not via deduction from more fundamental assumptions, but via abstraction from reality.

That brings us at last to the justification of metaphysical premises, which, similarly, is not by deduction, but via abstraction.

In a knowledge statement P => Q, you can see that Q is justified by P. We do not care how P is justified, or if this is even the case. — alcontali

Again, this is nonsense. If we did not care how axioms were justified, there would have been no controversy over the parallel postulate (which there was from the beginning) or about the axiom of choice. It is because we do care about the truth of axioms that so much ink was expended on these issues.

Why do we care? Because mathematics is a science -- as one organized body of knowledge among many. So, we want its conclusions to be true. In fact, truth is a central issue in Goedel's work. The problem he exposed (which completely undercuts your position) is that there are true theorems that cannot be proven from fixed axiom sets. If mathematics did not deal with truth, this could not be the case. If "mathematical truth" were convertible with provability, this could not be the case. So, the axiomatic method does not, and cannot, provide us with an exhaustive inventory of mathematical truths. That means that it cannot be the foundation of mathematical truth as you seem to imply.

Further, if the truth of P is indeterminate, so is the truth of Q if its sole justification is P => Q. On your account, mathematics is no more that a game -- not any different from Dungeons and Dragons, which also has rules that are neither true nor false, but simply to be followed by those playing the game. Funding mathematical research would be a scam in which we are paying people to play arbitrary games, with no hope of advancing our knowledge of reality, however theoretical.

Finally, it mathematics were not true, it would not be applicable to reality. Physicists who included mathematical premises in their reasoning, would be relying on claims of questionable or indeterminate truth, making their own conclusions and hypothetical predictions worthless.

to be continued ...