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.

Do you have a citation for Aristotle? — Dfpolis

No. Aristotle requires us to think if we are to understand him. Claiming that a being is the cause of being leaves unexplained the existence of that being. Here I am using explanation in the ordinary sense as Aristotle did. An explanation is discursive. Claiming that there is self-explaining being is not to provide a discursive explanation.

I agree that my argument uses insights due to Aristotle, ibn Sina and Aquinas. Still, being old is not a fallacy. Do you have an objection other than the ancient roots of my thought? — Dfpolis

I have no objections at all to the ancient roots of thought. I have made my objections clear. You simply posit what you cannot explain or demonstrate. It is just kicking the can.

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.

Has that caused you any difficulty? — Dfpolis

No, no difficulty at all. The difficulty is with your "proof".

Contingent facts cannot explain themselves. — Dfpolis

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.

I think you can work that out for yourself. The question is irrelevant to the soundness of my argument. — Dfpolis

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.

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

A great deal has been written about Aristotle's concealment. Here I want to point out a few things that may serve as hints as to whether Aristotle is arguing for the existence of God in the Metaphysics. Despite the appearance of writing as if these are things known, he gives us reason to doubt that assumption. This could be expanded and developed. It was done quickly and covers only a few sections of the text.

In Book Book 1, 2:

Hence also the possession of it [wisdom, universal knowledge of causes and principles] might be justly regarded as beyond human power.

He does not pursue this, however. There are some who suggest that Aristotle like Socrates was as skeptic, possessing human wisdom - knowledge of his ignorance, rather than divine knowledge

12:6:

… we must assert that it is necessary that there should be an eternal unmovable substance.

Why must it be asserted? Aristotle is a careful writer. If it is something known of could be demonstrated to be true then he would say so

For substances are the first of existing things, and if they are all destructible, all things are destructible.

Is there anything that we have knowledge of that is not destructible? If all things are destructible, as the available evidence indicates they are, then the assertion that it is necessary that there should be an eternal unmovable substance is questionable.

12:10:

Further, in virtue of what the numbers, or the soul and the body, or in general the form and the thing, are one-of this no one tells us anything; nor can any one tell, unless he says, as we do, that the mover makes them one.

Of course what one says and what has been demonstrated to be true is not the same thing.

1) What does contradiction inhere in? — tim wood

I am not claiming that the contradictions can inhere in things as accidents, i.e.,as what Aristotle calls secondary beings. I am claiming that they cannot exist, and so do not exist. As they do not exist, they do not inhere, and as they do not inhere, they have no need of something to inhere in.

If you're asking how we can form/justify this judgement given that what we are talking about does not exist, I respond that the judgement is not based on any experience of non-existence (which is impossible), but on our experience of being. Everything we encounter exists, and this allows us to abstract a notion of existence. (I say "notion" because it is not a concept like other concepts.)

When we do so, we see that existing utterly excludes not existing, and so we grasp the ontological principle of non-contradiction (A putative thing cannot both be and not be at one and the same time in one and the same sense.) It is this principle that is applied here.

Time for you to define existence and being, or to save you some trouble, to correct mine. Allow me to make a division into two classes: mental reality and extra-mental reality. Seven, for example, is a mental reality and not an extra-mental reality, as are all numbers, truth, justice, love, and the American way. — tim wood

I answered your definition question in the OP.

Lovecraft's theology does not begin and end with Cthulhu. The ultimate God in his pantheon is Azathoth, the blind idiot God.

Look around you.

Plausible, no? — Theologian

Perhaps Lovecraft derived this idea from the Gnostic's own "idiot" creator God, Yaldabaoth.

"In one of the ironies of mythic history, Yahweh himself became guilty of self-deification. In the book of Isaiah, the Jewish deity declares: “I am God and there is no other!” (46:9). In gnostic sources, this declaration becomes the mantra of the foolish creator, Yaldabaoth. This chapter examines three versions of Yaldabaoth’s myth (all found in Nag Hammadi codex II) in (1) The Apocryphon of John, (2) The Nature of the Rulers, and (3) The Origin of the World. It is argued that Gnostic Christians created the character of Yaldabaoth not to subvert Judaism itself but to criticize fellow Christians who adopted Yahweh’s superiority. By fitting the Jewish deity into the typology of self-deification, gnostics showed how foolish it was to believe in a jealous god who tried to prevent the deification of others."

From here: https://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780190467166.001.0001/acprof-9780190467166-chapter-4

Perhaps Lovecraft derived this idea from the Gnostic's own "idiot" creator God, Yaldabaoth. — Janus

I'm not enough of a Lovecraft scholar to say (actually not a Lovecraft scholar at all :wink: ). Although he was quite erudite, I'm also not entirely sure how many of the sources your article references would have been available to him.

It's also important to be aware that in describing Azathoth as a "blind idiot God" Lovecraft was telling us (as he more explicitly explains elsewhere, I think) that Azathoth was not sentient. So not quite a "blind idiot" in the usual sense.

"outside the ordered universe [is] that amorphous blight of nethermost confusion which blasphemes and bubbles at the center of all infinity—the boundless daemon sultan Azathoth, whose name no lips dare speak aloud, and who gnaws hungrily in inconceivable, unlighted chambers beyond time and space amidst the muffled, maddening beating of vile drums and the thin monotonous whine of accursed flutes."

~The Dream Quest of Unknown Kadath

Right, a quite different conception then, given that Yaldabaoth would seem to be thought of as both sentient and sapient, but deluded.

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 ...

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. — Dfpolis

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

Perhaps the best-known objection to a coherence theory of truth is Bertrand Russell's. He maintained that since both a belief and its negation will, individually, cohere with at least one set of beliefs, this means that contradictory beliefs can be shown to be true according to coherence theory, and therefore that the theory cannot work. However, what most coherence theorists are concerned with is not all possible beliefs, but the set of beliefs that people actually hold. The main problem for a coherence theory of truth, then, is how to specify just this particular set, given that the truth of which beliefs are actually held can only be determined by means of coherence.

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

Recall that the root meaning of "geometry" is "land measure" and many of its axioms are true of real-world geometric relations. — Dfpolis

You would have to visit all possible planets in the universe in order to verify that they are true of real-world geometric relations. You cannot do that. You will only sample some of these. Therefore, you cannot exclude the existence of counterexamples. Hence, these axioms are neither provable nor true about the universe.

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. — Dfpolis

Entanglement allows for simultaneous being and not being in the real world. Schrödinger's cat is another example. Therefore, nuclear physicists seem to beg to disagree with you.

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. — Dfpolis

You cannot visit all possible such angles in the real, physical world. Therefore, the theorem is not provable about the real, physical world. It is only provable in the abstract, Platonic world in which the provability of this theorem is the result of the construction logic of that abstract, Platonic world. You can perfectly-well visit all such angles in an abstract, Platonic world. That doesn't cost energy. In the real, physical world, you would need more energy that you could ever practically amass.

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. — Dfpolis

These mental constructs are abstract, Platonic worlds. They are not real. They are called Platonic because they are very similar to Plato's forms (but not necessarily the same):

Mathematical Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. The term Platonism is used because such a view is seen to parallel Plato's Theory of Forms and a "World of Ideas" (Greek: eidos (εἶδος)) described in Plato's allegory of the cave: the everyday world can only imperfectly approximate an unchanging, ultimate reality.

A major question considered in mathematical Platonism is: Precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, that is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One proposed answer is the Ultimate Ensemble, a theory that postulates that all structures that exist mathematically also exist physically in their own universe.

Platonism is the dominant philosophical view in mathematics.

Historically, most axioms have been abstracted from our experience of reality. — Dfpolis

Originally, yes.

Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.[1][2][3] Two of the most highly abstract areas of modern mathematics are category theory and model theory.

However, this is no longer the dominant source of inspiration for axiomatization.

For example, the lambda calculus has absolutely no origins in the real, physical world. Nor do the various combinator calculi. None of Stephen Kleene's work, such as his famous closure, have any origin in the real world.

The entire discipline of computability has no connection, and has never had any connection, with the real, physical world. There is absolutely nothing that looks like a shift-reduce parser in the real world.

The mathematical foundations of computer science have never been about mimicking the real, physical world. That would simply be an exercise in futility. A running process on a computer system creates a virtual world of which the nature is studied using Platonic abstractions. Suggesting that these virtual worlds have originally been abstracted away from the real, physical world, is absurd. They do not exist in the real, physical world.

Alan Turing's Halting problem is provable in the abstract, Platonic world of running processes. What is the link with the real, physical world? In what way does the real, physical world contain running processes? Where are the naturally-occurring CPUs and computer systems?

Seriously, mathematics transcends the real, physical world. Physicists are just one group of its users. I do not understand why they think that they would be so privileged in connection with mathematics? Historically, there used to be an empirical link, but that link has been abstracted away a long time ago. There is no 20th century mathematics that still has such link. Mathematicians do not desire such link, because it would hold things back. Such link is very, very retrograde. Ever since the axiomatization of set theory in 1905 by Zermelo and Fränckel, absolutely nobody still wants that link.

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. — Dfpolis

Well, ... in systems of which the associated language is capable of expressing the axioms of arithmetic.

Gödel was talking about the minimum power of a virtual machine and what we would today call its bytecode instructions. If the bytecode language can express Dedekind-Peano, the language can express (logical) truths that are not provable in the system.

Gödel's incompleteness is a language problem. The language required to express the axioms is more powerful than strictly what the axioms express. It is this fundamental mismatch that causes the problem.

In fact, Tarski's undefinability theorem is much better at expressing what Gödel's conundrum entails:

Smullyan (1991, 2001) has argued forcefully that Tarski's undefinability theorem deserves much of the attention garnered by Gödel's incompleteness theorems. That the latter theorems have much to say about all of mathematics and more controversially, about a range of philosophical issues (e.g., Lucas 1961) is less than evident. Tarski's theorem, on the other hand, is not directly about mathematics but about the inherent limitations of any formal language sufficiently expressive to be of real interest. Such languages are necessarily capable of enough self-reference for the diagonal lemma to apply to them. The broader philosophical import of Tarski's theorem is more strikingly evident.

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

It does work for morality. According to Kant's Critique of Practical Reason, the core of a moral system are its categorical imperatives, i.e. its axioms. In fact, Socrates already suggested that: "The understanding of mathematics is necessary for a sound grasp of ethics."

If they cannot be tested, they are unfalsifiable hypotheses and highly suspect. — Dfpolis

That is the empirical view in science, but a constructivist heresy in mathematics:

In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. Even though most mathematicians do not accept the constructivist's thesis that only mathematics done based on constructive methods is sound, constructive methods are increasingly of interest on non-ideological grounds.

If your only tool is a hammer, then the entire world will end up looking like a nail. If you cannot transcend the sandbox of physics, you will misunderstand and mismanage everything you ever do, outside physics. Is it so unthinkable to you that other epistemic methods are different from your own? Doing mathematics in the way you suggest, is simply not mathematics. It would be a failed form of physics.

As we have just agreed, the original justification of mathematical axioms was not via deduction from more fundamental assumptions, but via abstraction from reality. — Dfpolis

Only pre-20th century mathematics mostly originated via abstraction from reality. However, most of the progress that has been booked after that, does not.

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. — Dfpolis

Gödel does not talk about correspondence-theory "true". You can even trivially understand that from his canonical example:

S = "S is not provable in theory T"

Is S provable in T? No, because that would be a contradiction. Hence, S is (logically) true. Therefore, we are now sitting on a theorem that is (logically) true but not provable.

The language L associated with T is powerful enough to express S, and therefore, S is a relevant theorem in T.

The undefinability theorem does not prevent truth in one theory from being defined in a stronger theory. For example, the set of (codes for) formulas of first-order Peano arithmetic that are true in N is definable by a formula in second order arithmetic. Similarly, the set of true formulas of the standard model of second order arithmetic (or n-th order arithmetic for any n) can be defined by a formula in first-order ZFC.

In other words, it will be "true" and not provable in T, but it will be provable in any theory of which T is a sub-theory. The real, physical world is not chained into this tower of theories. Hence, it has nothing to do with correspondence-theory "true".

Further, if the truth of P is indeterminate, so is the truth of Q if its sole justification is P => Q. — Dfpolis

Yes, mathematical theorems are not correspondence-theory "true". They are only provable in their abstract, Platonic world.

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. — Dfpolis

There are no mathematical truths. There are only theorems provable from the construction logic of their abstract, Platonic world, i.e. their axioms.

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. — Dfpolis

Agreed.

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. — Dfpolis

There is no hope of advancing our knowledge of reality through mathematics. In relation to theories about the real, physical world, mathematics only supplies a consistency-maintaining bureaucracy of formalisms. Physics uses these formalisms. Hence, mathematics is useful to physics.

Finally, it mathematics were not true, it would not be applicable to reality. — Dfpolis

Mathematics is not applicable to reality. You will have to use another discipline for that purpose. You may indeed encounter mathematics as a tool to maintain consistency in what this other discipline claims, but that does not mean that mathematics would say anything about the real world.

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 — Dfpolis

Physicists do not include mathematical premises in their reasoning. They only maintain consistency in their theories by using mathematics. That works like a charm.

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. — Dfpolis

Hardy already admitted exactly that, in 1940, in "A Mathematician's Apology". It is not a secret:

Hardy preferred his work to be considered pure mathematics, perhaps because of his detestation of war and the military uses to which mathematics had been applied. He made several statements similar to that in his Apology:

I have never done anything "useful".

No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.

Hardy regards as "pure" the kinds of mathematics that are independent of the physical world.

Or even:

We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not.

Furthermore, the low-hanging fruit is only moderately useful. It is rather some of the hard, abstract stuff, that initially looks useless, even for centuries, that will eventually turn out to be a real game changer. For example, Euler's centuries-old work on number theory, became a multi-trillion dollar business when Rivest-Shamir-Aldeman (RSA) kicked off the world of public-key cryptography.

Seriously, it is not the quick wins that make humanity progress.

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.

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).

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.

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 haven't heard back from you on this, so I am going to assume I have misunderstood your claim. I think where I am getting tripped up is when I read the phrase "all possible acts", I think a set of all possible actions such as "lifting a hand", "taking a step" or "creating a physical being ex nihilo". So when you say that an infinite being can engage in all possible acts, it seems obvious to me that this is wrong because an infinite being can only execute a small subset of the set of all possible acts (some of which can only be executed by finite beings).

However, you seem to be thinking of the "set of all possible acts" in a different way. In your previous response you seem to be indexing possibility to the being in question. So instead of talking about the set of all possible acts in total, you're talking about the sum of all possible acts for a being of type X. So to say that a particular infinite being can engage in all possible acts means that it can engage in that subset of possible acts that any infinite being could execute. Whereas a particular finite being could never engage in the set of all possible acts that any finite being could execute (e.g. a man with no hand could not raise his hand, etc.).

Just trying to understand you. Is that anywhere close?

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.

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.

I have been visited by God all of my life & God demonstrated to me the science he uses to control the universe while staying outside of time & space.

What we think of as the material universe & world is only actually an energy. And that energy is moving at a particular wavelength/speed which is the Primary Dimension of the universe.

We mortals measure the 3 dimensions of objects including the universe & that presents us with measurements in time & space, that we are always striving to find newer ways to work with & overcome; in order to improve our lives.
But the basic (primary) unseen dimension of the universe is its wavelength & that dimension dictates the measurements of what all the other dimensions will be within the universe.

So when God manipulates the wavelength of the universe & of himself he completely steps out of the dimensions & their measurements that we are all locked & bound tightly in.

And so we see a universe that is vast & unconquerable 'while God sees a holographic type 'insignificant illusion that can be put aside completely & manipulated in any way at all.


God has demonstrated this to me over & over & let me hold metallic magnetic type elements that alter the wavelength of any atoms coming near them.
God builds very real solid gates from these simple elements & when we walk through the gates our atoms have their wavelength instantly changed & we enter a parallel world with all new dimensions & measurements, that are totally different than the earth's dimensions. It is completely real & stunning to experience.

I've watched millions of beautiful people join forums over the years to describe the wonders God has shown them, & without fail every person was instantly mocked into silence by the ignorant people who infest every forum. And not one webmaster or vile mod god has ever stopped the people from being mocked into silence.
Every webmaster & mod ignorantly works to facilitate & uphold the mockery of anyone who has a wonderful tale to tell about their meetings or experiences with God etc. In this way the web has always destroyed most all new knowledge the world of man has about God.
It does that by luring & destroying every new individual, as they arrive on demon infested sites to tell their wonderful story. It allows them to be insulted & bullied into silence.


So the good people of the world, & all the others, are without much new information about God because the web has destroyed/silenced all the people who tried to give that new information. Instead you are stuck with the old scriptures which cannot be removed so easily & yet have very little scientific type information about God.

Anyway I just found this forum a minute ago & haven't read any of your comments yet. But I read a bit of the opening post & it mentions how God can do everything mortals cannot do, & I just wanted to explain that God can do all those things because God has an extra dimension (direction/measurement) to work with, & that dimension is the speed of the wavelength that the earth & universe is made from.
By stepping aside from that wavelength God steps outside of space & time & can shrink all of space to zero & be anywhere in the universe at once; & stop time completely & casually walk around us, (at what to us would be the speed of light), & manipulate everything that happens in the world of man down to a T.
And we would experience it as miracles & impossible coincidences etc when it is actually just a science God is using to step outside of time so that he can manipulate everything that happens inside of time.

God has demonstrated vast amounts of his powers to me & I know they are true. God shows me the future thousands of times & it always comes true even when I think it is not possible for such things to happen.
So I know God is real & from real life experience with God I know that he simply uses science to control time & space & everything that is locked inside them. It is all so very simple to God & it could be simple to our own scientists too when they figure out how to manipulate the wavelength of the energy that the earth & universe are made from

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.

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?

There are lots of things I can do - that can be done - with a real brick that cannot at all be done with the idea of a brick. We're not going to get stuck on this are we?

Infinite being can effect any possible act either directly, or by indirection. — Dfpolis

Ah, I see. That helps clarify things for me.

It strikes me that the only possible act that God engages in directly is the act of creation ex nihilo. If this is true, then 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.

Are there any other direct actions God can take besides creation ex nihilo? If so, what are some examples?

because it is not possible to instantiate a contradiction. — Dfpolis

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?

But the notion of "contradiction," in ordinary usage is reasonably clear and plain. But is it so clear in a general sense? What is "contradictory' cannot be the same as the possible and not-possible, beacuse the latter is mutable, changes over time. It leaves the question, asked earlier, of the nature - the essence - of contradiction. 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.

In any case, we've devolved this notion of "God" from an omnipotent and infinite being to one who cannot do anything! Or at least anything that we would call doing. (I like ham, but can you do pastrami?)

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). — 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?

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. — Dfpolis

Do you imagine that when you close your eyes the world disappears? I have done just what you say must be done, only with your eyes shut you do not see it.

Then you should not claim the authority of Aristotle. — Dfpolis

Here is where we differ. I do not claim the authority of Aristotle. My position is that Aristotle is a zetetic skeptic - he inquires based on the knowledge that he does not know. You are a Christian who accepts what you have been told.

This mischaracterizes the argument. — Dfpolis

The argument is tortuous bending itself into unnatural positions in the hopes of escaping what is plainly evident. Aquinas' supreme being is a being, it is, it exists. As you say: God is "a particular, infinite being".

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. If you follow Aquinas you cannot. If you cannot explain the existence of God, you have not proven the existence of God. Positing an infinite being is not a proof of the existence of that being. Claiming that being is infinite and self-explaining is not proof that there is an infinite being that is self-explaining. Every being that is is what it is.

I made it clear in the OP that I was not talking about discursive explanations, but about dynamical ones. — Dfpolis

More equivocation. A proof is a demonstration. If it is an argument then it is by definition discursive. 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.

Premise 2: Whatever exists is either finite or infinite. — Dfpolis

We know that finite things exist but do you prove that an infinite being does?

Premise 6: A finite being cannot explain its own existence. — Dfpolis

In defense of this you claim:

But, being human does not imply that I exist. If it did, no human could cease existing. — Dfpolis

The finitude of our being is part of what it means for us to be.

Our existence is dependent. It is "explained" by the existence of our human and non-human ancestors, the earth, the sun, molecules, atoms, the fundamental forces. You might argue that each of these is limited, and that may be, but they are also capable of acting in such a way as to give rise to us.

That there is something rather than nothing may be both the starting point and limit of human understanding, but of course this does not satisfy the desire for a God who creates ex nihilo, a desire that conflates itself with the desire to know, a desire that confuses the dependence of individual beings for a dependence of all being save the being on which all is dependent, a being you declare is not dependent.


Added: There is a Jewish tradition that takes the dot in the first word of Genesis בְּרֵאשִׁית (read right to left) as symbolic of our inability to go back before when God began to create. We start, as he did, with a world that is chaotic, without form and with nothing distinct from anything else until God began to separate.

God has shown me many things & one day I saw God sitting on a high place creating universes. Each one is like the cell of a honeycomb that fits into all the others & yet is self contained. I asked God how he was creating the universes & he described that he uses the light that people call "God" to will them into existence, & they are all replicas of the other. God would create a universe & then we would go into it & explore it for a while, & then God would repeat the process.

Our own science thinks that in one sense everything is created from light. Yogis are said to be able to hold a crystal glass filled with water in their hands & the light of their own aura cumulates in the water & creates a mini universe in there.
The Vedas state that God lives in water & His aura creates the universes naturally inside the water.
The Bible states that God created the light first of all & then created a space inside water to put the universe in. And water remained above the universe when creation was finished.
So it is a bubble of light energy under water. And I have personally watched God creating universes from light & he explained how he does it & then took me into those universes.

No I don't expect you to understand or believe. hohoho

I simply want to dispel another notion that this thread states as fact. The notion that God cannot do the simple things that people can do etc. That is totally false. God can do absolutely everything we can do & infinite more things. I have held God's hand & walked with God many times. We eat & drink together & enjoy the sights & journeys we go on.

You need to comprehend a simple part of life first. You are a soul in a body & the soul is like an orb of intelligent electric energy that has no physical senses & yet it can experience what the mortal senses are experiencing when it is inside a body. As a spiritual soul you can do nothing on earth but when you are in a body you can do anything & experience everything through that body. That is what God does also. He utilises various bodies to manifest in & experiences & does everything through those bodies. So God is just the same as you or I & can do anything we can do & more. We fall unconscious for a moment when we die & yet God leaves his body in full consciousness & enters a different body. So God can do everything we can do & more. I often see God carrying a body suit around in preparation to use it to manifest in the earth realm.

There is a far deeper aspect to God & the soul that I haven't mentioned. The Spirit of God & our own souls come from bodies that are vastly superior to earth bodies. God has taken me to see it all & I have met countless people that God also showed these things to. Our souls & God's soul enter bodies & live lives in the material world & the other levels of the heavens & do & experience all things in all those worlds; & yet all of our souls are from vastly superior bodies which are waiting back at our source.

So I know from decades of vast experience that God is not limited in any way that this thread is trying to put forth. I understand your logic but it is based on incomplete knowledge & understandings & that has led you astray. You'll need to meet God in person to know more about God.

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. — Dfpolis

The theory (Q) is justified by its experimental test reports (P). Therefore, P => Q.

The universal claim is not justified by visiting all cases in the universe. It is justified by visiting a mere sample. That is also why Q is not provable, as a counterexample cannot be excluded. That is the essence of the scientific method.

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. — Dfpolis

Mathematics has its own canons. Science is one epistemic method and mathematics is another. Seriously, if your only tool is a hammer, then the entire world will soon start looking like a nail.

our capacity to investigate those foundations shows that being an axiom does not preclude justification — Dfpolis

That would only lead to infinite regress. Therefore, this approach is rejected in mathematics. As Aristotle said: "If nothing is assumed, then nothing can be concluded."

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. — Dfpolis

Maybe you should read the basic instructions of Oregon State University for freshmen novice students:

What "proof" means in everyday speech:

In casual conversations, most people use the word "proof" when they mean that there is indisputable evidence that supports an idea.

Scientists should be wary of using the term "proof". Science does not "prove" things. Science can and does provide evidence in favor of, or against, a particular idea. In science, proofs are possible only in the highly abstract world of mathematics.

What should scientists say instead of "proof"?

Scientists should use the term "evidence" instead of the word "proof". When we test our hypotheses, we obtain evidence that supports or rejects the hypotheses. We do not "prove" our hypotheses.

While this may seem like a subtle difference, the words we use can subconsciously color our thinking. "Proof" suggests that a matter is completely settled, that we have had the last word on something.
...
In this class, therefore, I will ask you all to be mindful of using the term evidence rather than proof.

You seem to have missed the very basic training that was supposed to teach you not to use the term "proof" outside axiomatic derivation in mathematics.

It looks very much like the Oregon State University would disqualify you, and bar you from calling yourself a scientist.

In fact, that is a generalized problem with scientism. The worse the scientific training, the more the person becomes prone to the problem:

Scientism is an ideology that promotes science as the only objective means by which society should determine normative and epistemological values. The term scientism is generally used critically, pointing to the cosmetic application of science in unwarranted situations not amenable to application of the scientific method or similar scientific standards.

As I have argued already, mathematics obviously has its own normative and epistemological values.

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. — Dfpolis

The statement "2+2=4" is trivially provable from Dedekind-Peano's axiomatization of number theory. Still, the fact that the statement is provable in the abstract, Platonic world of number theory does not necessarily make it correspondence-theory "true" in the real, physical world.

In fact, that is not even possible, because the numbers "2" and "4" are an abstract language objects that do not appear in the real, physical world. You can also call them "two" and "four", or "deux" and "quatre". These things are not real-world objects. They are language expressions. Since when do language expressions have physical attributes? How can something be part of the real, physical world without any physical attribute at all?

They merely work out the implications of axioms that may or may not be justified by our experience of the real world. — Dfpolis

The implications, i.e. theorems, are exclusively justified from necessarily following from the axioms. It has nothing to do with the real world. The axioms themselves are never justified. Otherwise, they would not be axioms, because their justifications would then instead be the axioms. That way of thinking obviously just leads to infinite regress. Hence, justification of axioms is a fruitless activity.

If the axioms are justified — Dfpolis

Justifying axioms is exactly what does not make sense for mere procedural reasons. If you must justify the axioms, why would exempt you from also justifying their justifications? That approach leads to infinite regress, and is therefore not viable.

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. — Dfpolis

The Stack Exchange question How does one justify funding for mathematics research? undoubtedly gives a reasonably good overview of why there is quite a bit of funding for reality-divorced, pure mathematical research.

One large and growing source of funding over the 20th century have been the military and intelligence departments. For example, you cannot do strong cryptography at any reasonable level without developing elaborate seemingly unrelated theorems in pure number theory.

In the link, you can see what kind of government departments and agencies subcontract research in mathematics. These grants are obviously not for plucking low-hanging fruit, such as slavishly mirroring reality.

Another funding source has been companies like IBM, who may rather be interested in fundamental computer science but often ends up dabbling in, and publishing pure mathematics. For example, Elsevier, a large academic publisher also has grants for research in mathematics.

Seriously, there is quite a bit of funding for reality-divorced research in pure mathematics.

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.

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.

Hello Dfpolis.

Good Op with some good reasoning.

The golden template of being having our lessor being coming from a greater Being is not what we observe hereabouts, for we note the ever more complex obtaining from the simpler and the simple, but there would seem to have to be something even behind the simplest—something eternal.

The template is also not impressive since it has to be discarded after only one usage, to avoid an infinite regress; however, this can be accomplished the way you have it, which is that the Being—albeit an assumption to have it be a person-like system of mind—is 'infinite'/'unlimited.

To have 'infinite'/'unlimited' to be substantial as a completed, finished state is troublesome, given that the 'infinite' cannot be capped as extant. There are dangers in using a word like 'infinite' as a stand-alone something or an amount/extent reachable, for its definition tends to some series or extent going on and on.

We also don't see that all was made instantly through an utmost power, but that the accumulations were long and slow, we barely making it through the great extinctions.

Perhaps I can add some support to what we might share as there having to be something eternal.

It would seem that the ultimate basis/existence needs be eternal, given that its supposed opposite, Nothing, cannot be. For those who might post about 'from Nothing', whatever appears from 'it' requires some capability/possibility/potential/random and so that would be the eternal something and so a total 'Nothing' was not had as claimed, bringing us back to an eternal. Perhaps the eternal is 'possibility', this needing only the same 'possibility' behind it.

It's still seems a bit troublesome for there to be an eternal something without its ever having been made, but with 'Nothing' out of the picture we have a mandatory eternal basis with no option, no choice; the eternal has to be; it must be. There is no selection or election.

What would the eternal be like? Well, how could it be anything specific/particular when there is no point before or outside it for it to be specified/designed? How does the eternal as something not at all particular be anything if it is so nebulous? Unknown, as rather formless and timeless, but, falling into assumption, might it have to be anything and everything?

Is the everything, then, all at once or little by little? We don't know the mode of time, whether it's of eternalism or presentism, so, we must profess our ignorance there, and other places, too.

What parallel do we have for something that is never anything particular? Well, the state of the universe never stops changing (precluding stillness), everything always transforming, even a trillion times a second, even if some semblances appear unchanging to our slow viewpoint. It can only be the eternal that is ever transforming (yet does so in a way that doesn't basically change it).

All in all, a 'maybe' is still a 'maybe', even if probabilistically unlikely, and so I'll give 'God' a generous 'maybe'.

Those who would preach either "God is, for sure!" or "God isn't, for sure!" are in line for being called 'misleading', and worse, 'dishonest'. Neither way can be honestly taught as truth.

That we can't know which is which, 'God' or not, appears to be the only truth to be gotten out of all this, but to some believers this would indicate that they cannot be blamed.

It all gets worse, due to fixed will, and that the supposed 'God' of the main religions can't be approved of, but those are other, secondary, stories, yet they hint that the possible making up of dogmas can run into grave contradictions.