Since 4=2+2, 2+2 and 4 are interchangeable. — jorndoe

I don't see how that's relevant. Since you and I are both human beings, we're interchangeable when someone says bring me a human being. It really means very little.



Sorry, I don't normally use Wikipedia and I only looked at the page on "equals sign".

Nevertheless, what is at issue is whether a so-called "mathematical object" is an object identifiable according to the law of identity. It is not, because two equal, but different things, such as the addition operation of 2+2, and the number 4 are said to be the same object. Therefore, despite what the Wikipedia quote indicates, and many mathematicians might claim, these two different things, the operation represented by "2+2", and the number represented by "4", cannot be "the same" if we adhere to the law of identity, which denies that two distinct things are the same object... The mathematical axioms which state that these two distinct things are the same thing are nothing more than deception. I know you'll continue in your denial, but so be it.

If time is what is emergent, then it is necessary that nothing be happening before it gets started. The idea of "before" becomes the incoherent claim here — apokrisis

Right, and "emergent" implies that there is a time prior to, hence a "before" the thing emerges. That's exactly the problem with your claim that time emerges, it necessitate such incoherencies as a time before time, due to its contradictory nature.

You presume time to be eternal. Thus there is always a "before". Hence time is proven to be eternal. Your argument is a simple tautology. — apokrisis

I never presumed time to be eternal. Why would you conclude that? I just argued that there is time prior to material existence.. How does that necessitate "time is eternal". I generally use "eternal" in the theological way, to indicate "outside time", so "eternal time" would be self-contradictory.

A thermal model of time is about the emergence of a global asymmetry - an arrow of time pointed from the now towards the after - the present towards the future. So the past, the before, is a backwards projection. It is imagining the arrow reversed. And reversed to negative infinity.

Yet the reality - according to science - is that time travel (in a backward direction) is unphysical. And the Big Bang was an origin point for a thermal arrow of time.

Yes, we can still ask where the heat to drive that great spatial expansion and thus create an arrow of time, a gradient of change, could have come from. What was "before" that?

But this is no longer a conventional notion of a temporal "before" anymore than it is a conventional notion of "what could have been hotter" than the Planck heat, or "shorter" than the Planck distance, or "slower" than the speed of light. — apokrisis

This is why theologians need the term "eternal" to refer to what is outside of time. The scientific community hijacks and restricts the use of "time" to conform to their empirical observations, i.e. they define time in relation to the material world. This leaves absolutely no way of talking about what is prior to the material world, because that creates the apparent contradiction of "before time". So the theologians use "eternal" to refer to outside of time, when "time" becomes defined in this scientistic way.

Every such conventional notion fuses at the Planck scale - the scale of physical unification. The asymmetries are turned back into a single collective symmetry. There is no longer a before, a shorter, a hotter, a slower. All such definite coordinates are lost in the symmetry of a logical vagueness. That to which the principle of non-contradiction (PNC) now fails to apply.

Before the PNC applied, there is a time when it didn't. That is the "before" here. — apokrisis

This is only the case from your scientistic perspective. If we ditch that scientism, and adopt some properly formulated metaphysical principles, as the theologians do, we can get out of that trap of having to assume that the PNC does not apply.

So for example, when you say "Every such conventional notion fuses at the Planck scale", you are restricting "conventional" to refer only to the understanding of time employed by the scientific community. All other ways of understanding time are excluded by your bias, as unconventional. So when the theologians demonstrate a way of avoiding such violation of the PNC by showing that we need to allow for activity which is outside of time (eternal actuality) when "time" is defined that scientistic way, your bias inclines you to dismiss it as unconventional, and an appeal to the supernatural.

The "Planck scale" only represents the level at which empirical observation becomes impossible. All you need to do is to accept the reality that there is activity which is impossible to observe empirically, to get beyond this self-imposed restriction. It is a self-imposed restriction, because you are restricting your reality (actual existence) to that which can be empirically observed. Once you open your mind to the truth that there is reality (actual existence) beyond that which can be observed empirically, this "Planck scale" restriction can be seen as unwarranted. And, the vagueness caused by that misconception of time can be properly dealt with through the application of logical principles such as the PNC. The inclination, or urge to violate the PNC is derived from the vagueness produced by that misconception of time. By denying the PNC you strip yourself of the capacity to understand what lies beyond what is empirically observable. It's a self-defeating metaphysics which you propose.

It is a claim of a theistic model. And a naturalistic model has become the one that has produced all the useful physics here. — apokrisis

That a model is useful does not necessitate the conclusion that it provides a true representation. Thales predicted a solar eclipse with an untrue model of the planetary motions. That the models produced by physics have encountered problems which incline you to say that the PNC is violated beyond the Planck scale, is clear evidence that they are untrue models, regardless of their predictive capacity. It's self-defeating to simply assert that reality is illogical beyond this level, so simply forget about trying to understand it. How could a reality which is illogical in its base level (the unobservable Planck level) produce a logically ordered upper level? Such a metaphysics is completely incoherent.

Epicycles to explain away a metaphysics that is provenly unphysical. It feels like an explanation being expanded but it is a confusion being compounded. — apokrisis

Evidence of your bias. If it's "unphysical", reject it. But all your metaphysics of "perfect symmetry" is equally "unphysical", so you're really just hypocritical. .

Do you understand what you mean by it (in your first sentence of the quote above)? Why do you think I mean anything different? You’re arguing against yourself here. — Luke

Sorry Luke, I have no Idea what you're talking about here. I did not use "it", and you're not making clear what "it" refers to. I said something different from what you said, so I think it's highly unlikely that we both meant the same thing. Since we did not say the same thing, it makes no sense to use "it" to refer to what both of us said.

I thought we were all talking about “the mathematical system”? — Luke

Just making sure that you understood this. You're statements appeared somewhat misleading, as if you were talking about "value" in an absolute sense, implying that numbers express the value of something in an absolute sense, rather than a specific type of value.

You're wrong mathematically and we are not getting anywhere. I tried to beg off the conv a while back but seem to be having difficulty executing on my intention. We're not making progress. I have nothing new to say. — fishfry

Why does Wikipedia agree with me if I am wrong? I can assure you that I didn't write the page.

If X = Y then X and Y necessarily refer to the same abstract object. There is no question about it. — fishfry

Why does Wikipedia disagree with you and say that under specific conditions the two things referred to have the same value, not that they are the same thing?

hen Thomas Jefferson wrote that "All men are created equal," he of course did NOT mean that they were mathematically equal, as 2 + 2 and 4 are equal; but rather equal under God and nature as human beings. — fishfry

Yes, two men are mathematically equal. One man is mathematically equal to another man. Each being "one" man, indicates that they are mathematically equal. Furthermore, each person, having the right to vote is mathematically equal to every other person, so they count the votes to see who wins the election. It is clearly the case that human beings are mathematically equal in a democracy, each having one vote. There is no other form of equality. "equal under God and nature as human beings" is just a fancy way of saying that under the condition of being human, we are all mathematically equal.

But it's not mathematical equality. I'd like to say it's beneath you to stoop to such a low rhetorical trick. But I guess it's not beneath you after all. Frankly it's beneath ME to have to explain this in words, it should be obvious. — fishfry

If you really think that two men are not mathematically equal to another two men, or five dogs are not mathematically equal to another five dogs, or an object of five kilograms is not mathematically equal to another object of five kilograms, or a five kilometer stretch of highway is not mathematically equal to another five kilometer stretch of highway, such that there is some other meaning to "mathematically equal", then tell me what that other meaning of "mathematically equal" is.

Because that's what mathematical equality is. That's how mathematicians define equality. Ultimately you have the same set on both sides of an equation. Once again you erroneously take your ignorance of mathematics as profundity in philosophy. — fishfry

Oh, now I see your problem. You think that because some set theory defines "mathematical equality" in this way, then this is what equality means in mathematics. As I explained to Banno, and to you already, this is a faulty definition, the one employed by the axiom of extensionality. It does not truthfully represent how "equal" is used in mathematics. It was devised for some other purpose, not for the purpose of representing how "equal" is used in mathematics. So, some set theorist defined "equal" in this completely faulty way, totally unrepresentative of how "equal" is actually used in mathematics, and you employ this definition as a premise in your argument against me. Your premise is a false premise, as evidenced by the Wikipedia quote. Your argument is unsound, being based in the false premise that mathematicians use "equal" in that way. They do not.

Yes, that’s why I said that both sides of the equation are different expressions of the same value. What part do you disagree with? — Luke

The different expressions represent different things with the same value. "2+2" says something different, it represents something different from what "4" represents, though we say that the two distinct things represented have the same value within the arithmetical system.' I don't really know what you would mean by "different expressions of the same value". That sounds like you are assigning value to the expressions themselves, rather than to the things represented by the expressions.

But it was not a representation of what I am arguing anyway, that's why I criticized it as a straw man. I am arguing that the different expressions are not different ways of representing the same thing, as in Banno's example of "Hesperus = Phosphorus". That is what fishfry is arguing, that you have the same thing represented on both sides of the equation. Even if we try to reduce "2+2", and "4" to being simple representations of "value", as you seem to want to do, we'd have to qualify what type of value. So we might say that they both represent "the same quantitative value", or "mathematical value". But we still cannot say that they have "the same value" in an absolute sense because there are all sorts of different value systems. That's why the Wikipedia quote refers to "the conditions under which they have the same value". "Value" is relative to a particular system of evaluation. So when it is said that "they have the same value", it is implied that they have the same value within a particular system of evaluation, the mathematical system.

I agree with that argument too. Which is why I say the matter of origination can only be solved by adding a logic of vagueness to our metaphysical tool kit.

Both formal and material cause have to arise in the same moment. They in fact must emerge as the two aspects of a shared symmetry breaking. And time (as spacetime) also emerges. — apokrisis

Again, you are ignoring the contradiction involved in "time emerges". Time must already be passing for anything to emerge, so time is necessarily prior to emergence. If space emerges, that's something different from time emerging. Time cannot emerge because that implies time is passing prior to time emerging, in order for this emergence to occur.

Your idea that formal and material cause must arise together, is the cause of vagueness in your metaphysics. You are not properly distinguishing the active from the passive. When form (as actual, active) is seen as prior to material existence, vagueness succumbs to absolution. Now we have time passing, with active Forms, prior to any material existence. This is the separate realm of non-spatial existence, described by dualism. So it makes no sense to describe this world of immaterial Forms in the spatial terms of physical matter. This is why quantum physics is so inadequate for understanding first principles of ontology, they speculate into the immaterial realm, using empirically derived principles of physics drawn from observations of material existence. One cannot apply the principles of material existence to the immaterial realm, they are distinct. The immaterial is separate and distinct from the material in the very same way that the future is separate and distinct from the past. It is only by denying this separation that vagueness is allowed to enter your metaphysics.

What we understand, in a mysticism based metaphysics, is that the entire material universe is created anew with each passing moment of time. This is a necessary conclusion derived from the nature of freewill. The freewill has the power to interfere with the continuity of material existence at any moment in time. This indicates that there is no necessary continuity of material existence between past and future. So all material existence must be created anew, from the Forms, at each moment of time. The human will, as final cause has the power to co-determine the material existence which will occur at each moment. The only vagueness is within the lack of understanding which the human mind has.

Thanks jgill, the others just don't seem to get it. We could say the same thing about all forms of measurement. Two different things are two kilograms, or two meters long, so they would be equal in respect to the particular system of evaluation. Likewise, "2+2" and "4" are equal in respect to quantitative value, but this does not mean that the same thing is signified by each of these two, only that they are equal in that particular system of evaluation.

You're arguing that both sides of an equation have equal (the same) value, but that they are different expressions of that value? Sounds reasonable. — Luke

What are you talking about? How can you be so daft in your interpretation, when I've explicitly stated over and over again what I am arguing. Your misinterpretation appears to be completely intentional, an intentional straw man.

I am arguing that what is represented by the right side of the equation has "the same value" as what is represented by the left side of the equation. But we cannot conclude that what is represented by the right side of the equation is "the same" as what is represented by the left side of the equation, if we adhere to the law of identity in defining what "the same" means. Do you understand the difference between "having the same value", and "being the same"?

I think you understand what I am saying, so don't bother replying with another straw man, fake interpretation. But if you have a relevant objection then I'd be glad to see it..

Do you have a source where it is clear that is the argument?

The Stanford article I cited on the prime matter issue fits with my view that Aristotle never fully worked it out, even if he left us with most of the essential tools. — apokrisis

The Stanford article doesn't seem to address Bks 9-12. This is where Aristotle "worked it out". You really just need to read his Metaphysics thoroughly from start to finish, to understand. There is another section, around Bk. 3 where it is demonstrate that in an absolute sense, form is prior to matter. This earlier argument, which is very similar to the principle of sufficient reason, along with the so-called cosmological argument are key.to understanding Aristotle's metaphysics.

The question he asks at this earlier point in the Metaphysics, is why does a thing, as a thing, exist as the thing which it is, and not something else. The thing, when it comes into existence, must come into existence as the thing it is, or else it would be something other than the thing it is, contrary to the law of identity. Also, a thing is not just random matter, it has a particular form as a particular thing. The only way that the thing could come into existence as the thing it is, and not some random other thing, is that it's material existence is preceded by its form. Therefore the form of a thing is necessarily prior to its material existence. And when we extend this to the universe as a whole, being an identifiable thing, this is the principle which supports Neo-Platonic independent Forms, which are temporally prior to the material existence of the universe. And, the fact that the form of a thing is prior to its material existence is what supports the human capacity of prediction, and free will to interfere in material continuity.

I agree with the first part but not the second. In my semiotic view, time as a continuous thread of Being is also emergent. — apokrisis

We've been through this before, this "view" is contradictory. Emergence is a temporal process, something occurring in time, as time passes, so it is impossible that time is emergent.

But physics has kept marching on until matter and void, space and time, etc, are all unified as aspects of a universal substance - a theory of quantum gravity, if we can pull that off. — apokrisis

Such a unified theory is not forthcoming, and the reason is that your contradictory view of time expressed above, is as you describe, the one accepted by physics. In other words, physics proceeds with a misconception of time.

Is this your interpretation? I don’t think he had the mission of refuting idealism as even Plato is not really an idealist - especially by the Timaeus. — apokrisis

We need to respect the evolving thought of Plato. He started out with a strict adherence to Pythagorean idealism. In this way, he laid out for understanding, the principles, such as the theory of participation, which support it. When this idealism was laid out, he pointed out its problems. Then he proceeded towards a new form of idealism, dependent on "God", which was expressed by the time he wrote Timaeus. Notice the similarity in this style, and what I described of Aristotle. They both lay out the principles, to be well understood, then they proceed toward criticism of them, and onward toward proposing something new. You cannot say that Plato is not idealist at this point, just because he rejected Pythagorean idealism for a new form of idealism, more similar to what is described by Berkeley.

That section of Metaphysics, Bk.9, ch.8-9, is clearly directed against Pythagorean idealism, and what Aristotle referred to as "some Platonists", as a refutation of that type of idealism. But Plato had already exposed the deficiency of the theory of participation, and that type of idealism, in The Republic, Parmenides, and other places I'm sure.

Instead I would say the issue was resolving the issue of hylomorphic substance - how substance could be the co-production of formal and material causality. Or as systems science would put it, bottom-up construction in interaction with top-down constraints. — apokrisis

You are totally neglecting what Aristotle demonstrated, the temporal priority of form, in relation to matter. This is what supports final cause, and our capacity to predict,.as well as interfere with through the means of free choice, what happens in the realm of material existence. Without accepting this principle, that form is prior in time to material existence (and this is the principle which necessitates the proposal of divinity), you cannot claim that your metaphysics is consistent with Aristotle's. You are removing the most important principle of Aristotelian metaphysics, because you desire to stay within the field of "naturalistic", instead of proceeding toward the supernatural, which is where Aristotle's metaphysics really leads us.

Yes. That is why I wanted to check how much scholasticism you are projecting onto what Aristotle actually says (as much as we can rely on the curated version passed down by history). — apokrisis

All you need to do is read Metaphysics Bk.12., specifically ch.7, where Aristotle explains how the first mover moves "the first heaven", which is the eternal circular motion. The first mover moves the first heaven in this way, because it is an apprehended "good". Therefore the cause of the heavens is a final cause, an intentional act. I've seen commentators, who like yourself are dissatisfied with Aristotle's reference to divinity, and propose that this part of the Metaphysics was not actually written by him. However, this is really nonsensical, because it is very consistent with the early part of Metaphysics, which I referred to above, where Aristotle questions why a thing is the thing which it is, and concludes that the form of a thing is temporally prior to its material existence.

What does the use of these terms have to do with the law of identity? — Luke

The law of identity is a fundamental law of logic, established to prevent deceptive use of terms. such as equivocation. We are discussing the use of terms within a logical system, therefore the law of identity is applicable.

But why can't these terms be used synonymously in relation to the law of identity? — Luke

Yes, "equal" and "the same" could be used synonymously. But if we adhere to "the same" as defined by the law of identity, and consider "equal" as it is used in mathematics, they are not synonyms. That's the point, this definition provides a false representation. That definition is a false premise. The equals sign is not used to indicate that the right and left side of the equation refer to the same thing, it is used as described in my quote from Wikipedia, to indicate that the right and left side have the same value. Therefore, to define "equal" as indicating "the same", for the sake of a logical argument concerning the nature of mathematical systems, or "mathematical objects", is to start with a false premise. Such arguments which take this definition as a premise are unsound because this is not the way "equal" is used in mathematics.

It appears to me like some people participating in this thread have seen "equal" defined in this faulty way, so according to that definition they conclude that "equal" is actually used in this way in mathematics. Do you see the equivocal sophistry here? Because it is defined in this way, they are misled into the conclusion that it is used in this way. They clearly have not taken any time to investigate, observe, and notice the reality of how "equal" is actually used in mathematics. If they had, they would see that mathematicians use "=" to indicate that two things have the same mathematical (quantitative) value, not to indicate that the things designated by the two sides of the equation are the same thing. So when these people produce a logical argument about the nature of mathematics, or "mathematical objects" and start with this definition as a premise, their arguments are completely unsound, being based in that false premise, which is not representative of mathematics.

I explained to fishfry already, that if the two sides of an equation actually referred to the very same thing, equations would be totally useless in application. To resolve problems, equations are used to compare distinct situations, knowns are compared with unknowns, producing information about the unknown situation through comparison with the known. If both were exactly the same, such a comparison would be unnecessary because we would know that the supposed unknown is really exactly the same as the known, and therefore really known as the same. The equation would give us no new information.

Atoms are theoretical to the core, a hypothetical image generated by arbitrary graphing systems to supplement the conceptualizing of dimensionless quantitative data, altogether assisting pattern recognition and prediction. I think the nature of matter depends completely on an observer's frame of reference. An essential disjunct exists between matter interpreted microscopically vs. macroscopically, and at the most basic levels there are multiple models associated with differing experimental contexts. As far as "inertia", it has not yet been possible to generate conditions in which matter is motionless, so I'm not sure how apropos this idea is unless referring to some kind of fundamental relativity. I would define matter as a relatively equilibrated state emergent from substance interactions. — Enrique

I interpret then, that you are saying that whether a state is equilibrated or not is completely dependent on one's perspective, one's frame of reference. You say the nature of matter depends on one's frame of reference, and you define "matter" as a "relatively equilibrate state".

So the question is, "relative" to what? Let's say that from a specific frame of reference, a particular state appears to be an equilibrate state. How would such a judgement be made, the state would appear to be equilibrate in relation to what? We couldn't judge it in relation to the frame of reference because we couldn't assume that the frame of reference was equilibrated. By what principle could we judge that there is any such thing as matter? Or is matter simply imaginary?

SO, for example, it would generally be accepted that Hesperus = Phosphorus. Would you accept that? — Banno

I already went through this. If two symbols refer to the same thing, they are equal. But it is a fallacy of affirming the consequent to say that if two things are equal they are therefore the same.

Well, thing is... most folk will disagree. — Banno

Did you read my quote from Wikipedia?

"The equals sign or equality sign, =, is a mathematical symbol used to indicate equality. It was invented in 1557 by Robert Recorde. In an equation, the equals sign is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value." — Metaphysician Undercover

Since Wikipedia agrees with me. it seems highly unlikely that most folk would disagree with me. Notice how "same" is qualified with "value". The right and the left side of the equation have "the same value". Anyone who knows how to read English knows that this does not mean that they are the same.

Well, thing is... most folk will disagree. — Banno

Really, you think that two equal people, because they are equal, are the same person. I think you are very wrong to think that most people would disagree with me. I think that there is a very small, and distinct culture of people who claim that equal things are necessarily the same thing. You might happen to be a part of this culture, and you like to believe that most folk are like you, to make it feel like your culture is more important than it really is.

But the terms "equal" and "the same" can also be used synonymously. Is this the basis of your dispute? — Luke

My argument is that this use, to use "equal" and "the same" synonymously, is in violation of the law of identity, and therefore unsuitable for any system of logic.

2+2 is the same as 4, — Banno

it's quite obvious, that in no way is 2+2 the same as 4. That's the point. It's very clear that 2+2 is equal to 4, meaning that the two have the same value within some value system, but at the same time it is also very clear that these two equal things are not the same thing. That is the way that "equal" is normally used, to refer to two distinct things of the same value. We never say that because two things are equal, they are therefore the same thing.

If you are defining "equal" to mean "the same as", as fishfry does, this is an unacceptable definition for a logical system because it is not consistent with the law of identity

So if you follow the directions and complete the process, then 2+2 = 4? — Luke

Of course, 2+2=4, and this is an acceptable statement. As I said already, not even a fool would deny that. What is at question is whether "2+2" and "4" both refer to the same thing. In other words, is there a difference between being equal and being the same.

Surely what he wanted to refute was an efficient first cause to the Cosmos. And this led him to claim that the actuality of Being must therefore be eternal.

So he got something wrong. We now know our Universe started in a Big Bang. There is a data point to be dealt with. — apokrisis

What he was actually refuting was Pythagorean idealism (Platonic idealism in modern terms). What he showed was that if ideas preexist human minds, their nature is as potential. The human mind is what gives them actual existence, in the act of "discovery". Then he demonstrated with the cosmological argument, that it is impossible for any potential to be eternal. This effectively refutes Platonic realism which holds the reality of eternal ideas.

The issue with respect to "matter" is that matter is itself just an idea. This might be hard for you to grasp, because "matter" is exactly what we assign to the physical world as what is independent from us, and therefore not an idea. But as "matter", is simply how we represent the physical world. It is our idea of temporal continuity, what persists unchanged in time, represented in science as inertia, mass, energy, etc.. In reality, what exists independent from us is changing forms, and we represent the aspects which are consistent, constant, as "matter", and this is the basis of the temporal continuity which is called "Being",

When he supposedly refuted idealism, by denying that potential could be eternal, he also refuted materialism, because materialism is actually just a twisted form of idealism, substantiated by the concept of "matter". And the concept of "matter" is not properly supported by empirical observations of the physical world, so it is deficient. The temporal continuity of existence, or Being, ends up being inconsistent with the concept of "matter" demonstrating that "matter" is just an idea, so materialism ends up being an idealism based in an assumed infallibility of the concept "matter".

Denying both materialism and idealism sets up the conditions for an eternal chain of efficient causes, sometimes called infinite regress. Aristotle referred to this as eternal circular motion, and you'll find a similar concept in the Hartle-Hawking no-boundary proposal. This eternal infinite regress is logically repugnant for a number of reasons, best demonstrated by the absurdities produced by the principle of plenitude which dictates that in an infinite amount of time, all possibilities have been actualized. So we move to an alternative first cause, which is "final cause", and the ensuing teleological nature of the universe.

To insist that the universe started with a Big Bang really doesn't get us anywhere. The "Big Bang" only represents the time in which our physical representations are no longer applicable. So the fact that we cannot understand the universe prior to this time called "The Big Bang", only indicates that out concept of "matter", or the modern representation, "energy", is deficient.

But his own theory of substance include finality - a prime mover. And if you put aside the suggestions that “God did it”, then his contrast of immobile celestial spheres and an actuality that is thus driven in circular motion Is not too bad a stab at some kind of naturalistic resolution. It is a fact of quantum theory that spin exists as a fundamental degree of freedom because the classical spacetime universe provides the motionless reference frame that makes it so. — apokrisis

The problem here is that we cannot get an acceptable "naturalistic resolution". This is because our conceptions of temporal continuity, "matter", or energy", are deficient. Once we get beyond this bias, this prejudice, that our conceptions of matter (or energy) are sufficient to give us a true understanding of temporal existence, Being, we see the need to look to other sources. The prejudice makes us believe in the infallibility of these conceptions. Recognizing this prejudice might lead us into the mysticism of human experience, will, intention, and free choice, as an alternative source for knowledge concerning temporal continuity. And when we properly understand the reality of intention, final cause, we cannot put aside the suggestion "God did it".

Here we have divergent courses of study. You would say that we ought to put aside this notion "God did it", stick with the demonstrably deficient and faulty scientific conceptions of temporal continuity, and ignore the vast wealth of accumulated theological knowledge of this subject. Thus you adhere to that prejudice which assumes a "naturalistic resolution" is possible, regardless of the mounting evidence against this possibility. On the other hand, we can take Aristotle's lead and proceed toward understanding the teleological nature of the universe, discovering the completely different understanding of temporal continuity, Being, which is explored in Neo-Platonism and early Christian theology.

An efficient cause is only so if it is efficient. And a fluctuation is defined by being a difference that doesn’t make a difference. Or only the weakest imaginable difference. — apokrisis

To say that there is a difference which makes no difference is either blatant contradiction, or to take a subjective perspective. If the former, then you take your ontology into dialectical materialism, allowing all sorts of confusion due to disavowing the law of non-contradiction. This sort of ontology is clear evidence of the deficiency in the concept of "matter". If the latter, then there's no point in speaking of the pre-human universe in such terms. There's a difference, but it doesn't make a difference to you, simply because you'd prefer to ignore this difference because it's evidence against your ontology. What kind of metaphysics is that? To claim that there is a difference which doesn't make a difference, simply because if it did make a difference it would be evidence against your ontology.

So for example we have this in the Stanford article I cited... — apokrisis

Sure, Aristotle goes to great extent defining "matter" in his Physics, and describing the concept of "prime matter" in the early part of his Metaphysics. This is to elucidate, and give a clear understanding of what "prime matter" is meant to represent, by those who assume such a thing. Then he proceeds in the later part of Metaphysics to refute this idea. That is his technique, to first elucidate the idea, so that we understand what is meant by the terminology, then he proceeds to demonstrate the deficiencies of the idea.

Fishfry claimed to have given me a proof of this, in the other thread, and continues to refer to this proof.

Do you accept this definition, that to be equal is to be the same? Are you and I the same, just because we're equal? Oh yeah, I remember now, you have no respect for the law of identity either, and you equivocate with "the same" in your interpretation of Wittgenstein's so-called private language argument. You think that if two distinct instances of sensation are similar, they can be said to be "the same" in the way that a chair remains the same chair if no one switches it out when you're not looking.

Do you understand the reason why Aristotle formulated and stated the law of identity in the way that he did? He did this to have a tool to be applied against such logical sophistry. When "the same" is used in a way which is not consistent with the law of identity, then we are not talking about objects. Objects are identifiable according to the law of identity. So this culture of mathematicians who use "same" in this way, and then proceed to talk about the mathematical "objects" represented, are engaged in this form of sophistry which we might call deception.

If we take jorndoe's suggestion, and say that numerals represent abstract quantities, we see very clearly that "4" represents one abstract quantity, and "2+2" represents two distinct abstract quantities with a mathematical operation of addition represented. If we replace "abstract quantity" with "object", there is no rule which dictates that "2+2" could represent one object. So this culture, which assumes that "2+2" represents an object, which is the same object that is represented by "4", just because two plus two is equal to four, is a culture of sophistry and deception.

Trump desperately wants the riots to continue, because he hasn't anything else going on than the "Law & Order" thing. — ssu

Yeah, Trump seems to like to manufacture these imaginary crisis situations at election time. I seem to remember something about a caravan of immigrants at the midterm. It's rather pathetic that he ignores the real crisis and manufactures an imaginary one.

After months of arguing this point, fishfry still seems to think that the equals sign means the same. Next, fishfry will offer me a proof of this. As you can see, we've been through it before.

and as I've argued before, there is this confusion between prime matter and primary substance - between the primacy of whatever could constitute the material aspect of hylomorphically-emergent actuality, and primacy that is then the actualised or enformed being which is thus the substantial substrate of further change and development. — apokrisis

This is what Aristotle claims to refute with the "cosmological argument", the idea of "emergent actuality". This is why he needed to posit the "prime mover", rather than the infinite potential of Anaxamander's apeiron. Later in his Metaphysics, the prime mover is described as a divine thinking.

There is what I would consider to be prime matter as Peircean firstness or vagueness. — apokrisis

So Peircean firstness, and the metaphysics which follows from it, is not at all consistent with Aristotle's metaphysics, because it adopts the very principle which Aristotle claims to have refuted. You really can't just overlook the fact that Aristotle replaced the concept of "prime matter" with "prime mover", as the foundation of his ontology, to pretend that Peircean metaphysics is consistent with Aristotle's. I mean, that's a pretty significant difference.

But when we talk of becoming preceeding being, we mean the Anaximander's apeiron or Peirce's tychism - potential as the pure spontaneity of unformed material fluctuation. If we had to describe such a general grounding to Being, it would be a materiality with the least possible substantiality. And even then, we should be imagining it as just naked "becoming" as "prime matter" with any materiality has already crossed that threshold into the realm of actualised Being. — apokrisis

This is exactly the notion of pure potential which Aristotle claims to have refuted. I don't see how you cross that threshold into the realm of actualised Being.

The equal sign means that the two sides are equal, just as "equal" indicates. Here's what Wikipedia says:
"The equals sign or equality sign, =, is a mathematical symbol used to indicate equality. It was invented in 1557 by Robert Recorde. In an equation, the equals sign is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value."

Clearly you are wrong to say that it's everyone's rule, that the equal sign means that the right and left sides refer to the same object. This rule is an expression of your idiosyncrasy.

I already went through your converse error, but I'll explain it to you again, as you don't seem to get it for some reason. I believe the formal fallacy is called affirming the consequent. If two symbols refer to the same thing, then there is necessarily equality between what the symbols refer to. But this does not mean that two equal things are the same thing. Do you understand this so far? Many things are equal, like two human beings, two dogs, or two cats, in the sense that the two distinct things can be given the same value. A human being might be equal to a dog if the evaluation criteria is being an animal. And do you see that the equal sign means that the right and left side are equal, as the Wikipedia articles says? How can you conclude, without the fallacy of affirming the consequent, that two equal things are necessarily the same thing?

The Danish Neils Bohr and German Max Planck, along with contributions from many additional scientists, successfully theorized matter as a duality appearing more particlelike or more wavelike depending on experimental context. — Enrique

This says to me, that "matter" might appear as a particle, or it might appear as a wave, depending on one's perspective. But do you see the need to define "matter". Is "matter" to you, the temporal continuity of existence, what apokrisis called "inertia"?

Wave and particle concepts were combined in a theory of all energized mass as ‘quantized’, occurring in discrete bundles that are however spatially diffuse in ways still, in the 21st century, only probabilistically definable. — Enrique

I think there may be a problem with equating "matter" and "mass". Depending on how you might define :"matter" it might not be necessary that matter has mass. But wouldn't this make a mess of Newton's laws?

Quantized ‘wavicles’ were found to be arranged within atoms... — Enrique

No one found any "wavicles". This is theory, which you are presenting as empirical evidence.

I acknowledge that nature does not at all mirror the double slit experiment, but matter has intrinsic wavelength as far as I know, correct me if that isn't accurate. — Enrique

What would you mean by "matter" here?

Read specifically Bk.9, Ch.8, where he explains how actuality is prior to potency, and how anything eternal must be actual, not potential. This excludes the possibility of prime matter as described earlier as an eternal, and first potential.

Obviously, therefore, the substance or form is actuality. According to this argument, then, it is obvious that actuality is prior in substantial being to potency; and as we have said, one actuality always precedes another in time right back to the actuality of the eternal prime mover.

But actuality is prior in a stricter sense also; for eternal things are prior in substance to perishable things, and no eternal thing exists potentially. The reason is this... — Aristotle, Metaphysics 1050b

Read Metaphysics book nine.

Those versed in quantum mechanics find it likely that extremely rapid rate of turnover in the ion flow cycle of nerve cells necessitates that these ions take the form of a tunneling wavicle as they enter and leave cells through transport channels. — Enrique

You ought to realize that a "wavicle" is not a real thing.

Surely what Aristotle meant by prime matter is one of the most fraught debates in metaphysics. But it can’t be cashed out as mental stuff. Nor even, immaterial essence.

It is more like a fluctuation or the least possible notion of a material action or efficient cause, in my view.

Peircean Firstness or tychism in other words. — apokrisis

Aristotle discussed the concept of prime matter, because it was a common speculation in his time. He ended up proving that it is impossible that such a thing as prime matter, or infinite potential, is real, with his cosmological argument. Therefore, prime matter is taken by Aristotle to be a fiction.

They do refer to the same abstract object, — fishfry

We're just going around in the same circle. By what principle do you say that this string of symbols "2+2" refers to one object? How would you distinguish which symbols refer to an object, on their own, and which objects require a string of symbols to represent them? Is it your rule that the left side of an equation refers to one object, and the right side of an equation refers to the same object?

The "means" or "process" of getting to the abstract object is irrelevant. — fishfry

It's not irrelevant because "2+2'" clearly refers to an operation, or ":process". There is an operation symbol "+", within the phrase. This is the fact you are ignoring, and are in complete denial of. How can you deny this fact? It's right there in black and white, "+". You look at "2+2" and read it as symbolizing the abstract object "4", refusing to interpret what the symbols actually represent. If this is not a very clear and obvious example of misinterpretation, then how else can you explain your ignorance of what is written? Do you actually recognize what is written but have some reason to deny it?

But I don't think of this as a process leading to 4. I think of 4, as the primary object, that already incorporates all of the processes that could lead to it. 2 + 2, 3.9999..., etc. — fishfry

Do you not see that this is a false way of looking at an effect? If there exists an object, and there is a multiplicity of processes which may have led to the existence of that object, only one of those processes is the correct process which caused the object. It is false to say that the object "incorporates all of the processes that could lead to it". This misconception leads to many ontological problems, because it allows that infinite possibilities, and infinite chains of processes inhere within an object, rendering an object as inherently unintelligible. To employ a system of mathematics which makes the existence of objects unintelligible, by incorporating infinity into the object's existence is self-defeating, if one's goal is to understand the existence of objects.

The causes and processes are secondary to the essential existence of the abstract number 4. — fishfry

I can agree with you on this point, the causes and processes are secondary. But are you ready to agree that since there is a "+" in "2+2",this phrase refers to a process?

But again you're just repeating your confusion. The number 4 incorporates within it 2 + 2 cats or 3.999... or whatever. They all point to the same thing. They're not "ways of getting to" the thing. I can't imagine why you have such a strange idea. — fishfry

Again, you are ignoring the significance of "+". Please read the phrase, and interpret it according to what the symbols actually say, not according to some preconceived notion of what you want the symbols to say.

Sheboygan, Wisconsin is the same identical city whether you get there from Milwaukee or Green Bay. — fishfry

Right, now if you were in Green Bay, and followed the directions of how to get from Milwaukee to Sheboygan, you would not get there from Green Bay. Likewise, if you were at 6, and followed the directions of how to get to 4 from 2, i.e. "+2", you would not get to 4 from 6, following those directions.

This is your problem, you are reading "2+2" as 4, instead of reading it as directions of how to get to 4. .

Are you identifying an object such as the number 4 with the process that "creates" it? I think that's a pretty big stretch — fishfry

No I'm not making such an identity. But I'm saying that if you assert that "2+2" is the same object as "4" you are claiming such an identity. The process of adding two with two will make four, but it is not four. I am saying that the process which makes four is not the same as the number four itself. A cause is not the same thing as its effect. You are insisting that the two are the same.

The number 4 is the number 4, and it's inherent in its nature that it can be represented many different ways. — fishfry

I really can't believe that you do not see the difference between what "2+2" represents and what "4" represents. The former indicates two quantities of two, with an operation of addition also indicated. The latter indicates one quantity of four. To interpret "2+2" as representing the number four is very clearly a misinterpretation.

The proof is that we can represent the relationship between two distinct 2s in many different ways, such as "I have 2 dogs, and 2 cats", "2,2", "2+2", "2X2", etc. In all such instances of representing two distinct quantities of two, they must be interpreted as two distinct quantities, to avoid misinterpretation. If you do not follow this simple rule of interpretation you completely disregard the application problem of adding apples and oranges. If two groups of two are automatically four there is no way to avoid the category mistake described as adding apples and oranges. In other words, you do not allow any provision for the reason why they were represented as distinct in the first place. Therefore, in insisting that "2+2" represents the same thing as "4", you are denying any valid reason for representing the two 2s as distinct in the first place, and nullifying that representation, of two distinct 2s, as an invalid representation.

Abstract quantities (is the phrase) — jorndoe

A number such as what is indicated by the numeral "4", is an abstract quantity. In the case of "2+2" , there are two distinct abstract quantities signified with "2" and "2", along with an operation signified with "+". Therefore it is incorrect to say that "4" represents the very same abstract quantity as "2+2", because clearly there are two distinct abstract quantities signified by "2+2", related to each other by the signified operation "+", not one quantity.

If I had known colleagues who were concerned about Platonic ideals, irrational numbers, or transfinite set theory I might have more to offer, but those issues were at best peripheral to our interests. — jgill

We all have our own interests. Since one of mine is philosophy, and mathematics really is not, I approach this issue from the perspective of being concerned about Platonic ideals rather than the associated mathematics. Many others in the forum will say to me, forget about what Platonic ideals are, or whether "Platonic ideals" is a valid concept, take the axioms for granted, and discuss the mathematics. But if the substance of the axioms is Platonic ideals, I want to understand the validation.

You guys still chatting about whether 2 + 2 = 4 or 2 + 2 ≠ 4 ? — jorndoe

There is no question that 2+2=4. Not even a fool would deny that. The question is whether "2+2" represents the same object as "4".

You are looking behind the symbols to the mathematics they represent. They are certainly equal in this regard. But if you look superficially at the compound symbol "2+2" and the singular symbol "4" as ink squiggles on paper, they clearly are not the same. But, of course, that's not your perspective. I am simply giving an instance when two things are equal in one sense, but not the same in another sense. You and fishfry can argue ad infinitum it seems. — jgill

I think it's true that fishfry and I will never agree, but the disagreement between you and I appears to be as to what constitutes a "mathematical object", or "Platonic ideal". Mathematicians may have defined "object" in such a way that "2+2" represents the very same object as "4", but I disagree with this definition, as it does not properly represent what a Platonic ideal really is. So I believe that the definition was manufactured for the purpose of doing what the mathematicians wanted to do with it, rather than with the purpose of representing what a Platonic ideal really is. That is not an acceptable way of doing logic, to manufacture premises which will support the desired conclusion. It is a case of petitio principii

The problem is, as Aristotle demonstrated, that there is an inherent and fundamental incompatibility between a process (becoming), and an object (being). The object is intelligible, and the process is fundamentally unintelligible. So a symbol like "+", or any type of operator, or function, does not represent an object, and represents something which is fundamentally unintelligible. And if we allow this element of unintelligibility into the object, to say for instance that "2+2" represents an object, then the object itself becomes fundamentally unintelligible. This might be the Kantian perspective, to represent the noumena (intelligible object), as fundamentally unintelligible, but it is not the Platonic perspective, and you can see how it is inherently contradictory to say that the intelligible object is fundamentally unintelligible.

In other words, if the Platonic ideal is supposed to be an eternal Form, meaning a truth which escapes from the corruption of time, it cannot be represented as containing within itself a temporal process. So these representations of Platonic mathematical objects, which allow processes (functions) to inhere within the object, are not proper representations. They are not proper because any such process, which inheres within an eternal object, must have infinite temporal extension, being a never ending, never changing infinite process, such as Aristotle's perfect circular motion. But this metaphysical idea has been demonstrated as a false one. Ideal perfection cannot be granted to a process.

,

Irrelevant. 4-ness is the ideal in discussion. — jgill

You've forgotten about summation. It might be the case that "4" represents 4-ness, but "2+2" represents a particular instance of the general rule of summation, not 4-ness.

Well maybe. I think that point's a stretch. Plato could be wrong. But more to the point, 4 already includes within itself the possibility of being partitioned into 2 + 2. or 1 + 1 + 2, or 1 + 1 + 1 + 1. This is in fact the mathematical subject of partitions. It's what Ramanujan was working on inThe Man Who Knew Infinity. IMO doing a good job of explaining the partition function to a general Hollywood audience is one of the greatest math feats in cinematic history.

Point being that if 4 is an "ideal" or whatever you call it by itself, it ALREADY CONTAINS the possibility of all its positive integer partitions.

Truly, 2 + 2 and 4 are the same Platonic object. I don't find your argument convincing for this reason:

Sure, 2 + 2 expresses the fact that 2 and + can be combined to make 4. But 4 already expresses the fact that 4 can be represented as 2 + 2. Partitions are a natural and built-in aspect of a number.

Am I at least representing your position correctly? — fishfry

I think you almost understand, but not quite. The symbol "4" represents a particular unity, if we adhere to Platonic idealism. Four is an object, a number. It may be the case that this number could be constructed through the summation process of 1+1+1+1, but this process is not the same thing as the object itself. Processes are not objects, they are activities which objects are engaged in. A cause is not the same as the effect. So it is ontologically incorrect to say that this process which creates the object "4" is the same thing as the object "4".

Evidence of this fact is that the object "4" may be created in an infinity of other ways, "6-2" for example. So, all these logical possibilities which are inherent within "4", cannot be the same as "4" because each one is itself a different process. The fact that a different process can be utilized to make an object indicates that the process is not the same thing as the object.

How can they represent the same Platonic ideal when "+" represents an ideal in itself, which is part of "2+2", but not part of "4"?

If we acknowledge that the term 'morally right' is applied to different behaviours/characteristics for different reasons in different contexts, we can see that no such mechanism can possibly exist. — Isaac

The fact that different people use the same word in different ways does not necessitate the conclusion that there is not a correct way to use it. It could be that they have just not learned the correct way. So your cited evidence supports a probabilistic conclusion, but it does not have the strength for your claimed conclusion "that no such mechanism can possibly exist".

Morality, as a single measurable property of behaviours/characteristics is a fabrication of philosophy, it just doesn't exist among real human groups. — Isaac

The issue here is one of judging the particular instance as suitable to be classed as a member of a specified general category. If a "real human group" (whatever "real" means in this context) has agreement amongst themselves, that a designated particular is a member of a specified category, how is this a fabrication of philosophy rather than a real attribute of that real group? I would think that if there was disagreement between two "real human groups", and a compromise was arbitrated by a philosopher, this would be a fabrication of philosophy. But what makes you think that it is philosophy rather than "real human nature", or something completely arbitrary, rather than philosophically directed arbitration, which produces such conventions?

In other words, why do you believe that a particular being a member of a general category (x is of the type A, for example), is something created rather than a natural fact? You seem to give no credence to the reality of types. But is it not true that there is a real difference in type between a human being and a chimpanzee, for example?

PA is a formal symbolic system no different in principle than the game of chess. — fishfry

That's not true, we went through this in the other thread. The application of the rules of mathematics always has a different purpose from the application of the rules of chess. If we want to apply mathematics toward understanding the universe, we need truth (in the sense of correspondence) in the axioms. Playing chess has a purpose of competing with another person for supremacy, within extremely limited conditions. If the rules of the game are designed to fulfill the purpose of the game, the people who have composed the rules must have kept that purpose in mind when creating the game. If not, there would be incoherence within the rules of the game, having been designed for different purposes.

If utility is removed from mathematical axioms, and they are composed simply for aesthetic beauty, then there will be random difference between one set of axioms and another, and real application would not be practical. This is not what we have in mathematics. Therefore we can conclude that mathematical axioms are not composed for aesthetic beauty.

If the purpose of mathematical axioms is simply utility, in an unconditional sense, then different mathematical systems will be inconsistent with others, depending on the purpose (the game) they are designed for. This means incoherency within the mathematical rules as a whole. This is what do we have in mathematics, as you yourself admit.

To produce consistency within the rules of mathematics as a whole, there needs to be one principle of utility, one purpose for which all the axioms are designed. Since mathematics is most widely applied in sciences we can look at the purpose of the axioms within science to get an idea of what that fundamental purpose might be. There are two distinct purposes which appear to me, one is the understanding of the universe, as mentioned above, and the other is for the prediction of events.

These two purposes are distinct. The former involves the bivalent logic of truth and falsity, while the latter involves probability. There is a fundamental incompatibility between these two, expressed in the law of excluded middle. When the axioms of one meet with the axioms of the other, paradox appears, as demonstrated by Zeno. This is because the logic of being (what is and is not) is inconsistent with the logic of becoming (what will be). The arduous task of the ontologist (metaphysician) is to determine the principles by which the two might be related to each other, to establish compatibility between them.

Note that as usual I ask you direct, probing questions and you'll respond by changing the subject. — fishfry

I'm changing the subject because your analogy has been demonstrated as completely insufficient and not irrelevant. Playing chess has a completely different purpose from applying mathematics.

There is no middle 'e' in judgment. Jus' sayin' but nevermind . Axioms are formal statements, strings of symbols that are well-formed according to specific syntactic rules. — fishfry

An axiom is a proposition which may or may not be accepted. That is the nature of an axiom. There are various reason why one might accept or reject a proposition. To say that an axiom must follow "specific syntactic rules" is one of these reasons, but clearly there are others.

Therefore there can be no "truth" in axioms; only logical consistency and interestingness. — fishfry

Obviously you have considered the acceptance and rejection of axioms from a very narrow perspective, without observance of the many real factors involved in this process.

Depends on the contexts of usage. — jgill

The attempt to unnecessarily restrict usage is fraught with problems. We are creatures of habit, and if habitual interpretation is different from the one imposed by a logical rule, the habit will often impose itself into interpretation of the logical conclusion in the form of equivocation.

So for example, we can define "equal" as "same", such that within this logical circle all uses of "equal" mean nothing other than "same", and all uses of "same" mean nothing other than "equal", but there would be absolutely no point to this. We could just use one of those words without losing anything. The only reason to use both, is if they have different intension. But that difference in intension requires that we determine the relationship between the two before we proceed with any logic procedure.

If in common usage, "equal" is the broader category, such that not all cases of being "equal" mean "the same", and all cases of being "the same" mean being "equal", then we ought to adhere to this in our logical definitions and proceedings to avoid possible confusion and equivocation. So there is a fundamental principle which we habitually recognize, and this is that two distinct things might be equal, but only one thing is the same as itself (law of identity). This distinguishes the uniqueness of a "thing" from any other similar or equal thing. But we commonly use "same" to refer to two distinct things which are of the same class, type, or category, and this is a completely different meaning of "same". If we equate "equal" with "same", we appeal to this other meaning of "same", which allows that two distinct things are the same. But then we are in violation of the law of identity.

It's not that bad! — fdrake

If agreeing with me was really that bad, I'd be afraid to leave my house.

No, I'm making a claim of Peano arithmetic, a purely syntactic system. — fishfry

This is exactly my point, such axioms are based in ontological principles, they are not "pure mathematics. You can insist that there is no ontology to them all that you want, refusing to consider the evidence, in denial, that's a matter of your own free choice.

You pointedly ignored my argument and wouldn't even engage with my having presented it. — fishfry

Actually I demonstrated your faulty interpretation of the premise of extensionality. That two symbols refer to something of "equal" value is not sufficient for the conclusion that they refer to "the same" thing. Being the same implies being equal, but being equal does not imply being the same. You commit a fallacy of conversion.

You are the one in denial, insisting that mathematical axioms are exempt from judgements of true and false, being "pure mathematics", and absolutely abstract, refusing to accept the truth in this matter.

No, rather it's everything and every type of thing. And to get back to the point; everything and every type of thing does not require a privileged "now" for it to exist. — Janus

That's a generalization, so it's not relevant unless you propose Platonic realism which allows for the non-temporal existence of universals. We're still at the same issue, you are assuming that concepts have existence independent from human minds.

As Kant pointed out it is only perception that requires time in the sense of a present moment. — Janus

This is where Kant might have gone a little off track. Conceptions are dependent on perception, so there is no such thing as a priori concepts. Aristotle made this argument against those Platonists and Pythagorean idealists who argued that position. Eternal Ideas are demonstrated as impossible.

It may well be that given Anscombe's particular usage of the word deterministic, her argument is logical and her conclusion sound

However, the general reader who believes that they know the common usage of the word deterministic may find her argument unclear.

In such a case, where the author uses a word in a way that is different to common usage, then the author should explain what they mean by the word at the beginning of their article. — RussellA

It's questionable whether using a word in an unusual way produces a sound argument. For the sake of a logical argument, one can define a word in any way the person wants. But a definition ought to be taken as a premise. And a false definition is a false premise.

This is true only if "utility" includes fascination with exploring a subject, finding what's behind the next intellectual door, where an investigation might lead, etc. That's been my motivation for many years. — jgill

Yes, utility includes that, because there's always a reason why one explores one subject rather than another. I study for the very same reason, to find what's behind the next intellectual door, but I don't deny that there's always a reason why I head in one direction rather than another.

No, that's a psychological claim — Gregory

Any claim, which anyone makes, says something about one's mind, so they can all be said to be psychological claims. So that's really irrelevant. But to say that "4" refers to a thing, and that "2+2" refers to the very same thing, rather than that "4" and "2+2" have meaning (in which case one might see that the meaning of each differs), is to make an ontological claim.

I am surprised you've heard of it. Yet you don't agree that 2 + 2 and 4 refer to the same thing. In my prior conversations with you, you've convinced me that you utterly reject symbolic mathematical formalisms. And without those, there certainly aren't any convergent infinite trigonometric series. Those are very abstract gadgets. There's a mismatch in your level of discourse. — fishfry

There's no mismatch in my discourse, you simply refuse to try and understand what I'm saying. I believe that two plus two equals four. I do not believe that "two plus two" and "four" refer to the same thing. Since you think that they refer to the same thing, you and I give "2+2=4" different meaning. We simply interpret this phrase differently. It is an ontological difference. So I reject some conclusions of mathematical formalism as unsound, based in unsound premises. This does not exclude me from taking a look at some of these unsound conclusions. Comparing unsound conclusions with what is really the case helps in the effort to produce better premises.

We keep coming back to the same point. Nobody is making metaphysical claims except you. I agree that SOME scientists think their theories are True with a capital T, but I don't. You're fighting against someone's opinion that isn't mine. — fishfry

The premises, axioms, theories, are metaphysical claims. whether you recognize this or not. I know we disagree on this, and you think that such premises might be based in something called "pure mathematics". but I explained to you in the other thread why this is an unsound principle itself. There is no such thing as "pure mathematics" in an absolute sense. Mathematics is ultimately guided by utility, and even those who might seem to be engaged in pure math are doing what they are doing (choosing whichever problems they choose to be working on instead of working on other problems) for a reason, so utility cannot be removed from mathematics.

Ok, fine. I stipulate that. Science isn't metaphysics, science is not ontology. What of it? I've been conceding you this point for days. You won't even acknowledge that I've said that, you just keep coming back with arguments as if I haven't said it. — fishfry

Do you recognize that scientists, in their scientific endeavours, regularly employ metaphysical principles?

Do you understand that I make no ontological or metaphysical claims for science? — fishfry

In saying that "2+2" and "4" refer to the very same thing, you make a metaphysical (ontological) claim.