Conclusion: I can't even begin my run, let alone finish it. I'm at the starting line and I'm stuck, I can make not an inch of progress. — Agent Smith

Convergence in calculus is thought to have long solved this. But before that, the classical solution was the Aristotelian solution in the form of potential and actual infinity. Some would say that it's inadequate.

Also, you can alternatively reject mereological descent which is a prerequisite for the argument, in the form of space and time, but some would say that spawns worse problems (i.e. Ibn Sina's distance function argument) with regards to discrete geometry and the success of physics.

I think you might have missed my point. If A is not A, then it can't equal A right? — Philosophim

Have you heard of the phrase "when pigs fly?" It is a adynaton, namely in that when it postulates a subjunction believed to take on a highly implausible (or impossible) premise to ridicule on whatever follows. A is A in any valuation of A, so A is not A is simply never true. But entertaining A is not A simply entails trivialism in classical FOL, where any proposition you want to follow follows. (This is well known as the principle of explosion).

You don't find some empirical evidence for why things aren't themselves. You're just forcing a proposition that's already taken to have no truth-conditions in FOL to somehow be true. It's incoherent.

"A" exists, and someone demonstrated to me that "A" did not exist, then A would be proven false. — Philosophim

A is just a placeholder. A unicorn is a unicorn. It doesn't matter if it exists or not. It's not an existential claim, it's an identity claim.

"A = A" and someone made it impossible for ~A to be a consideration — Philosophim

~A is considered in A=A. But A=A returns true even granting ~A. ~A is literally just a negative truth valuation for A. So in both cases if A is false then the other A is also false, and so on. Like I've shown you earlier in the truth table, you can value A with any combination of truth and false and it'll always be equivalent to itself. There's no way out of it.

"God exists and made the world" = A. If I said, "Could I attempt to show that something else created the world?" I would receive a response. If the person said, "Well yeah, I guess that's fine," I would then ask, "So what would be enough to show that God did not make the world? — Philosophim

It would be a counterexample to the proposition "God exists and made the world" because that proposition is not a tautology. But "God is God" or "Making the world is making the world" is a tautology that is always true regardless of whether God existed or not. In the same fashion that "Santa is Santa" is a tautology with no falsity conditions.

If they say, "God is beyond our understanding and definition," then there's really nothing to falsify. There's no definition or understanding of God to claim, so there is nothing to refute either. In short, non-falsifiable." — Philosophim

While this may be a claim you can't empirically falsify, it's not a tautology in logic. This is just beating a strawman.

No, they are very falsifiable. When would 6 not be 6? When 6=5 is one example. — Philosophim

"When" 6=5? There is no time where 6 is equal to 5. I'm actually appalled that we're debating such a simple notion. Simply asserting falsities with no truth conditions is not an argument. There are no conditions where 6=5 holds, so there isn't a time that you can reference where you say "when 6=5" because it's simply never the case. On the converse, 5=5 will always be true. You can literally manually check this if you don't believe me: that's what I gave you in my earlier response, which seems insufficient for your purposes

. It turns out that ~6=6 isn't true, but a contradiction. Therefore while we have a means of falsifying, we cannot show that 6=6 is false. Therefore, it must be true. — Philosophim

Conceding on trivialism still wouldn't falsify 6=6, it'd just make everything (including negations) trivially true.

Much appreciated, but we don't need it for what we're talking about as I think you can see from my examples above. — Philosophim

I'm fairly certain we do. You make very bold assertions with regards to the fields of logic and mathematics, but do not seem to grasp explanations of why these assertions are quite literally incoherent. I'm sure that further familiarity would not work against you and perhaps will lead you to the same conclusions I'm trying to communicate to you in this interaction.

I felt the need to tell you this, because I felt you did not understand falsifiability. I didn't take offense to your recommendation to read up on logic, don't take offense on me telling you things I don't think you understand either. — Philosophim

I'm sorry if you felt this way, but I want to be clear I was simply inquiring for the reason why you did so, i.e. what lead you to the impression I did not understand falsifiability. This is not the same as me taking offense. Generally, assume that I take no offense unless I indicate otherwise :).

Recall you just mentioned that you understood falsifiability was not the same as "impossibility" — Philosophim

If a proposition is impossible, it is necessarily false, whereas if a proposition is false it is not necessarily impossible.

All impossible propositions are false, but not all false propositions are impossible.

Again, those are both falsifiable statements. But, we cannot meet the requirements to show they are false. Therefore they are proven to be true. — Philosophim

You can't meet the requirements to show they are false because there are no such requirements. Propositional calculus is truth-functional, meaning the truth-value of a formula is a function of the semantics of the operators and the truth value of the propositions contained within it. All you're doing is exhausting all possible truth values. So tautologies return true having exhausted all possible truth values of false or true to all the propositions embedded within it. So there are no conditions where they're false.

1.If God exists, he would have created the best possible world.
2.There are cases where evil does not lead to the fruition of some greater good (ex: holocaust, starving children, etc.)
3.God could have created a world without these types evil
4.Therefore, God did not create the best possible world [2,3]
5. Therefore, God does not exist. [1,4] — tryhard

The Theodicy (Leibniz) response is basically P1 while rejecting P2. If God exists, he must have made the BPW, so any evils must entail some greater goods in the future. On a different avenue, a theist can employ chaos theory to undermine epistemic warrant for P2.

It seems that the problem of evil is the most powerful argument against the theist argument. — tryhard

Cantorian arguments, modal collapse arguments, and many more do a far better job

necessity means exists. None is supported — Gregory

To say that necessity entails existence is uncontroversial and does not need support IMO. If a being exists in all possible worlds it exists in the actual world, which is a possible world.

Incorrect.

If A=B, then a=a is false. — Philosophim

I'm not sure how familiar you are with logic, but this is pretty evidently untrue. I'll show you a truth-table if you don't believe me.

Notice how that in cases where a=b is true, a=a is true as well, and so the conjunction of both are true.
The only falsity conditions for the conjunction are when a=b does not hold, but notice how a=a remains a tautology regardless of the truth value of a=a



Logical equivalence is a transitive and symmetric relationship, so I'm not sure how you would even reach the conclusion that a=b would falsify a=a because you can just substitute the formulae around freely if they both hold. If not, then a=b simply doesn't hold.

Another example is probably from mathematical equality (which is not the same as logical equivalence, but a=a still is unfalsifiable even if = is understood as a mathematical equality and "a" as a variable). For example, 5=5 or 6=6 are still unfalsifiable truths. You can say 2+3 is equal to 5 too, so 2+3=5, but no one in their right mind would suggest that 5=5 is false because 2+3=5 is true.

I recommend this introductory course on logic from Stanford. In supplement, I'll also link this article explaining mathematical equality. I suggest that you familiarize yourself with these on your own freetime going onward with this conversation so that we have an easier time communicating.

Falsifiability does not mean, "It is necessary that it is false." It just means there can exist a condition in which it could potentially be false. An assertion must always allow the potential of its negation. — Philosophim

I'm aware that falsifiability is not the same as impossibility, rather it is simply possible falsity. I'm not sure why you felt the need to tell me this. Clearly, some propositions like a=a or some mathematical formulae like 5=5 have no falsifiability conditions and simply cannot be impossible. Similarly, in modal logic, the necessitation rule of K says that if some proposition is a theorem then it is necessary in all possible worlds, i.e. it's negation is logical impossibility, a fact that does not align with your view.

So I think you understand now. Physicalism is falsifiable by stating it could be the case that physicalism is false. — Philosophim

In the case of my example, the opponent of physicalism does not simply falsify physicalism but allow for its logical possibility, rather find an internal contradiction in physicalism. All contradictory sets of facts are logically impossible in any consistent modal logic, i.e. they simply could not be true. There isn't a world with square circles, or vice versa.

Now that you understand what falsifiability is, do you still have an objection to it? — Philosophim

I don't object to falsifiability in the context of the empirical sciences, where I believe it may even be beneficial. I think falsifiability as a philosophical or mathematical requirement is an incoherent position because both philosophy and mathematics have some facts that are given the status of being necessarily true and also unfalsifiable, like a=a or 5+5 and what not.

A very good question. First, it needs to be something falsifiable. — Philosophim

While falsifiability can definitely be proper of scientific discourse, for good reason even, I think it is seldom at all a good condition of philosophical or mathematical discourse which includes philosophical evidence that is sometimes given in the form of proofs. This is because some of the truths that philosophers and mathematicians deal with genuinely have no falsity conditions, i.e. all tautologies, like a=a or (p∨(q∧r))→((p∨q)∧(p∨r)), simply cannot be falsified but are undoubtedly true.

In a similar manner, contradictions are falsums, and in classical logic or other logics that uphold noncontradiction, if we have a contradictory formula like p∧¬p, then this always returns false whereas its negation ¬(p∧¬p) will be a tautology: i.e. will always return true and cannot be false, thus is unfalsifiable.

How does this relate to materialism or to philosophical discourse in general? Well, a common objection in philosophical argumentation is a self defeat objection. If an opponent of a position finds a contradiction in its doctrine, then if that contradiction is genuine, the doctrine will be always false. And so the negation of the doctrine will be always true with no falsity conditions.

In the context of the materialism/physicalism, the thesis that there exists only the physical, then if an opponent of the doctrine found it to be contradictory and was hypothetically successful, his proof of the negation of physicalism will be unfalsifiable by definition due to the logic outlined earlier. And this trivially entails the existence of at least one non-physical entity granting physicalism as false.

But it seems very unreasonable to dismiss a self defeat objection, which warrants at least one non-physical entity in the context of physicalism/materialism, in virtue of the fact that it's unfalsifiable. In philosophy, your opponents may think that your position is not just wrong, but literally could not be correct, so a well-motivated objection is oft unfalsifiable when successful (i.e. objections to the Christian Trinity as incoherent, if successful, are unfalsifiable, but are still nonetheless sound objections in these instances where they succeed).

For these purposes, I think falsifiability is a terrible criterion in the context of philosophy, but may be more fit for other uses like science or other empirical inquiry, and therefore also urge that you reconsider it.

We all know it. Time is unidirectional. — EugeneW

We actually don't. The very start of the post rules out C theory, which rejects temporal directionality. C theorists only agree with temporal order, and would tell you that the timelines (1) ABC and (2) CBA are identical in virtue of the fact that the betweenness relations (C-properties) of these timelines are sortally equivalent. Meanwhile, both the A-theorist and B-theorist would find there to be distinction between (1) and (2) through either A-properties or B-properties.

But neither of these are a default position that lacks a burden of proof, so the A and B theorist must be the ones motivating the A-properties or B-properties. And I'm not so sure any of the strategies used are adequate. Phenomenal or intuitive arguments used in an A-theoretic fashion can be perfectly rationalized by indexical accounts, and the issue I perceive is that the accounts B theorists use to undermine A theory are not sufficient (but necessary!) for B theory. In other words, undermining temporal dynamicity on its own is necessary for both B theory and C theory, but only sufficient for C theory.

Because of this, while I may not strongly commit to a C theoretic understanding of time I think it is a highly plausible one and one of our best explanations. So I would warn from presupposing it away in a discussion about time, and I'd definitely invite its insight.

For further reading, I recommend Matt Farr's paper On the adirectionality of time. It is excellent.

It depends on the religion. The anattā of Buddhism is philosophy in the most proper sense of the word, and this debate on the self has been going on between Buddhists and Hindus for a considerably long time. Similar concepts can be outlined in certain kinds of religions, like the example earlier, that are philosophical.

Meanwhile, Abrahamic religions should not be understood as philosophical any more than they ought be understood as scientific. Indeed, the Muslim philosopher Al Ghazali highlights this starkly in the Incoherence. Abrahamic religion makes claims on science without the empirical facts to substantiate it just as it makes claims on philosophy without the appropriate philosophical methodology to justify said claims. This is not an accusation against religion, nor me saying that religion is bad: this opinion is one you will find many theologians and theistic philosophers, of the Abrahamic tradition, agree with me on.

I want to reply to another comment here.

You can use all the philosophy in the world to bolster something, but if the core element of it is that it is "revelation" from the supernatural, and therefore "it cannot NOT be true" because of this, it can't really swim with the other philosophies because everything has to fit that supernatural revelation that cannot NOT be true. — schopenhauer1

This is hardly a good criticism. Tautologies cannot not be true, and I'm sure we wouldn't rule them out, so we definitely need to find another feature to critique aside "not being able to not be true," because we certainly accept things that cannot not be true like facts of logic and mathematics. Certain types of theists view God's existence in a different fashion: it is simply a base of their paradigm. Plantinga develops this in his reformed epistemology where belief in God is justified as properly basic and not the kind of belief that really needs justification. Meanwhile, William James, much prior to Plantinga, grounded belief in God in virtue of his pragmaticism oriented epistemology and truth theory.

Keep in mind I'm not arguing /for/ either Plantinga's position nor William James's. I'm simply expounding what I perceive to be an inadequacy in your criticism and using examples from the theistic side to demonstrate how people may formulate their worldview.

Throughout the history of philosophy, there has been two primary approaches to a more formal and technical understanding of (i.e. philosophical) omnipotence as depicted in theology.

These two approaches are the Thomistic approach & the Cartesian approach, respectively after Aquinas and Descartes.

• In the Thomistic approach, omnipotence is the ability to actualize any logical potential, or bring about any logical state of affairs. Impossibility & violating logic is not part of an omnipotent being's ability, because impossible potentials do not exist, and because (in Aquinas's theology), logic is of God's rational and orderly nature.
• Descartes thought otherwise. Thus, omnipotence, as conceptualized by Cartesians, is the ability to actualize any potential whatsoever with no qualification on being in accord with any system of logic. This means that unlike a Thomistic omnipotent, a Cartesian omnipotent can bring about square circles and so on.

Keep in mind that these interpretations are tools, and so it is not necessary that one interpretation is correct and the other isn't, rather, they're done such that philosophers can interact with this concept, and different religious faiths may see omnipotence in different ways. In other words, the "correct" interpretation is indexical to the theology at hand.

But regardless, there's another distinction that's important here, and it's whether there can be an omnipotent being such that its omnipotence is accidental to itself. In these cases, presumably, that being can survive removing its omnipotence, because its omnipotence is not an essential component of its identity.

However, in the cases of an essential omnipotent, to remove its omnipotence is to remove its essence and effectively kill itself. Notice that both the Thomistic & Cartesian interpretations of omnipotence (without God inserted) wouldn't confine this because no logical contradiction is explicitly evident. The issue is that Thomists argue that accidental omnipotence isn't a thing, and that omnipotence can only possibly be true of God, such that God is also necessary. This is because Thomists commit to a doctrine known as divine simplicity (DDS) that makes it so God is atomically simple and has no proper parts. What this entails is that all of God's attributes (omniscience, omnipotence, necessity, etc) are actually one attribute and it's simply our mind that fails to capture this unity when thinking about God.

So given DDS, there isn't such a thing as omniscient but not omnipotent, or omnipotent but not necessary. To say there's an omnipotent being is to also say there's an omniscient, necessary, etc being because all of these attributes in our language are different intensions that fixate the same extension. The interesting part about this is that since necessity is true of this being, then its non-existence would be a logical contradiction, so it can neither remove its omnipotence nor kill itself (a Thomist would say both of these are the same thing!)

However, while a Cartesian may admit of DDS, and may even come to agree with the Thomist that this self-destruction is actually a contradiction, a Cartesian would say an omnipotent can still bring it about because it is not bound by any laws of logic. The stark point of contention in the theology here is that a Cartesian may be inclined to see logic as something that might "rule over omnipotence," whereas Thomists understand logic itself as a result of the orderly & rational nature of that being which is omnipotent.

So ultimately, it depends on the type of omnipotence you're using. To a Thomist, no, but to a Cartesian, yes.

Hope this answers your question.

Materialism is a metaphysical position concerning ontology, and so it cannot be answered by empirical means in virtue of its very nature. So science cannot falsify or verify materialism: in fact, materialism is not a scientific hypothesis: it is a philosophical one.

This goes for idealism, dualism, et. cetera. None of these are the kind of theses that can truly interact with empirical investigation.

So yes, it is unscientific, as a lot of philosophy is (which isn't exactly a bad thing, either.)

Is there a mathematical and or logical expression for comparing the properties and lack of properties of Objects? — Josh Alfred

Yeah, Px implies some P is true of x. ~Px implies that it's not the case that some P is true of x.

Lack of properties will get you in some messy territory, like incomplete objects & quantified predication. Consider:

1) x is that which has the single property of being blue
2) But having one property is truly said of x
3) So x has two predicates!

But I'm sure this isn't what you meant, just food for thought because object completeness is a relatively interesting topic

But yes, pretty much all of predicate logic is dedicated to doing what you seem to be asking for

Oh my, this is what I really wanted to avoid. I didn't expect to meet something of this nature on this website, haha, so I will give you one response and you can make what you want out of it.

Silly is ad hominem name calling and not an argument — magritte

This is not what an ad hominem is. An ad hominem is assent to a negative doxastic attitude to a person's claim in virtue of nothing else aside the person's character or credibility: it is an (informal) fallacy because descriptions of the individual have a different truth-maker than their proposition, and even in cases when they coincidence (i.e. someone self-describing), the intensional context would be distinct even if the extension overlaps.

So, an example of an ad hominem is P1: X said "Y" P2: X is silly! C3: Y is false.

A common misconception is that anything that could be interpreted as somewhat insulting, be it to a person or to an argument, is an "ad hominem." Since my characterization of the claim as silly was entirely independent of the argument I provided against it, that literally cannot be ad hominem.

It only suggests that you lack familiarity with the subject, therefore you intend to tackle the opponent instead of the claim. — magritte

I'm really not sure if this is attempting to characterize /me/ or /my intention/, but in either case I don't think it's worthwhile to address since it seems predicated on something I just refuted.

Objective truth is a golden dogma and you can stick to it as you will. — magritte

This is a clear strawman of my position. I don't commit to "objective truth" nor do I stick to it, neither do I stick to "subjective truth." Instead, as I clarified very clearly in my initial comment, "objective" vs "subjective" has nothing to do with truth and that OP is meddling in an unnecessarily nonrelated topic, when he should be instead pursuing something more pressing to his concern like the epistemic possibility of mind-independent knowledge.

But others are at liberty to question the comfortable surroundings of strict and limiting non-contradiction with excluded middle. — magritte

I'm not sure how this at all has anything to do with what you or I said earlier. I do not care if the individual uses a non-explosive/explosive or indeterminate/determinate logic, in fact, little if none of what I said truly addresses or concerns itself with these axioms of LNC & LEM or their capacity to be questioned.

By 'literature' you just mean standard dogmatic literature taught to undergraduates. — magritte

You can psychoanalyze all day if it makes you feel better. I'm not an undergrad, and this is literally the philosophical literature on the topic nonetheless your perspective WRT this. It might be the case that my description of the literature on the topic isn't congruent with what you want it to be, hence the charged language of "dogmatic" and "undergrads," but I can't stress this enough- 'objective' and 'subjective truth' are pop culture topics that have exactly nothing to do with the various theories of truth proposed and this sort of emotional response doesn't change the facts of the matter or speak to what I said in any way.

It's OK, but there is much more to logic, and truth is only a value of a logical calculation in whichever logic one might choose. — magritte

This is an awful equivocation... something I could only chalk up to either intentional bad faith or complete unfamiliarity with the topic. Logic, as a discipline, is considered with consequence, which in other words translates to things like truth-preservation and thus fixating the behavior of truth. This does not overlap with the nature of truth that truth theories like correspondence, coherence, and what not attempt to answer. Truth-valuation is not the same as truth, and it is also why you can introduce truth predicates to logics that have truth-valuation but not truth predication (many very prevalent formal systems, like propositional calculus, don't actually have a truth predicate).

Different logics outline different behavior for truth, different theories of truth outline different candidate explanations for what truth /is/. You equivocated the latter with the former.

A combination of both these are known as the axiomatic theories of truth, which introduce truth predicates to base logics (which are still truth-evaluating) to further answer questions about the nature of truth in of itself: that's not the job of normal logics, which are only concerned with the truth of arguments (i.e. how the truth-valuation of propositions can lead to other propositions), which is why First Order Logic can be perfectly used by a correspondence theorist about truth or a pragmaticistic or any other theory. And this holds for the predominant majority of other logics, like Graham Priest's LP.

Nonetheless, I'd like to remind you as I said earlier that I will not bother engaging with this sort of activity any further, so whatever response you send if you choose to give one I will not reply to. If you are okay with this, you are free to respond further. This is just for transparency in your expectations of our communication to save you any potential disappointments, haha.

Do you really think Kant didn’t know about spherical geometry? And didn’t take care to qualify his postulates accordingly? — Mww

Kant died in around the early 1800's. Bolyai made the first publication of non-Euclidean geometry around three or so decades afterwards. Gauss had the same ideas for the majority of his life in drafted notes but never published them for fearing controversy.

And these were top mathematicians. I'd suspect Kant didn't somehow discover non-Euclidean geometry decades before the mathematicians did, let alone keep it in his head despite discovering it especially when it offered such a strong challenge against a central doctrine of his work.

Mathematicians before this time (aside Gauss) thought that only Euclid's axioms could consistently capture geometry, in other words, that Euclidean geometry is the only geometry. This (mistaken) idea is what set ground for one of Kant's important ideas, which was corrected briefly after his death.

So yes, I "really" think that. In fact, I don't see what's so surprising or unusual about thinking that people generally don't discover things that were discovered after their death, because had they, they'd be the ones who actually discovered them, and for the reasons provided earlier I can't imagine anyone seriously thinking Kant, instead of Gauss, Bolyai & Schweikart is who truly discovered non-Euclidean geometry, especially considering not only the lack of evidence but the severe implausibility that comes along this sort of claim.

All this talk about "objective truth" is silly. Objectivity and subjectivity are properties that pertain to a mind. None of the literature on the theories of truth (pragmaticism, coherentism, correspondence, semantic, et. cetera) ever seems to actually investigate this isolated pop-culture idea about "objective vs subjective" truth.

Perhaps you mean to speak about a different topic, whether there can be a true proposition known mind-independently, or something of that sort. That topic has to do with epistemology, and you're better off investigating the literature on disputes regarding empiricism, the possibiltiy of apriori knowledge, disputes regarding mind independence, and so on, rather than meddling in truth theory for an entirely unrelated and nonrelevant subject.

So then, what you'll have to do is describe how someone would concluded such a fact, moreso than simply stating as much. Which is to say, explain how your Gauss and Schweikart discoverd the complex nature of non-euclidean geometry without reference to any principles theretofore established by which to do so? — Garrett Travers

I'm a little confused by the nature of your request here. Axioms and first principles are first principles for a reason, namely that they're not a "conclusion," a sort of proposition entailed by some prior set of propositions. Because had they been conclusions, they'd simply be a theorem of some prior proposition instead of an axiom. And if these propositions they're entailed from aren't axioms, then you just run the theorem game forever until you reach the axioms: the stopping points.

Our subjunctive mood would be that given different definitions & axioms, we get different theorems and therefore geometries, so there's little reason to not abandon the synthetic thesis because the analyticity is conspicuous.

Yes. Several philosophers agree with you.

One, Descartes believes quantity is proper of physical multiplicities in the sense of extension:

8. A thing that has a certain quantity or number isn’t •really distinct from the quantity or number—all that’s involved is •distinctness of reason. [See 1:62.] There is no real difference between quantity and the extended substance that has the quantity; the two are merely distinct in reason, in the way that the number three is distinct from a trio of things. ·Here’s why they have a distinctness of reason·: Suppose there’s a corporeal substance that occupies a space of 10 ft3—-we can consider its entire nature without attending to its specific size, because we understand this nature to be exactly the same in the whole thing as in any part of it. Conversely, we can think of •the number ten, or •the continuous quantity 10 ft3 , without attending to this particular substance, because the concept of •the number ten is just the same in all the contexts where it is used, ten feet or ten men or ten anything; and although •the continuous quantity 10 ft3 is unintelligible without some extended substance that has that size, it can be understood apart from this particular substance. ·And here’s why they aren’t really distinct·: In reality it is impossible to take the tiniest amount from the quantity or extension without also removing just that much of the substance; and conversely it is impossible to remove the tiniest amount from the substance without taking away just that much of the quantity or extension. — Descartes

From the second part of Principles of Philosophy.

And in fact Avicenna had a similarly veined argument, a sort of identity between quantity and ostensibly the extension it "models," to forward against the atomists (not the same as the Greek tradition, rather, atomists only in the mereological sense) of his era. That is, because, Aristotle's critiques, which more primarily concern the broader metaphysical doctrine rather than the specific atomistic thesis, Epicurus had developed atomism a tad bit more and the Arabic Kalam tradition had numerous proponents of atomism (who were only considered with the thesis of a discrete space rather the Greek doctrine in full). Since they'd take the notion of discreteness to be a literal representation of space, Avicenna argued that the functionality of the Pythagorean Theorem in physical space, and its utter incompatibility with discreteness even at a level of approximation, disproved atomism.

Finally, in more recent literature, material numbers have been a charge utilized to dispense away various logicist accounts of the philosophy of mathematics. This is known as the "Julius Caesar problem," namely that in accounts where numbers are objects, the principles to which we designate numbers can't tell us when exactly something is or isn't a number. What I mean is, take something like Hume's Principle, we wouldn't know when to identify an arbitrary object, i.e. Julius Caesar, with a number.

The number of a material object is then a kind of measure of the built-in positionality of a material object.

However, this bit of your post, I don't understand. Perhaps instead of the very extension of the object itself, you're referring to spatial or temporal location, if I understand you correctly. If so, this is an example of an extrinsic, relational predicate (or an impure property). There is debate about whether these are truly proper to the objects they designate or not, but here, the quantity would moreso be a reference of things like physical distance (presumably "material"). The way this would be treated would depend on if the underlying metaphysics for space & time is absolute or a container, i.e. Platonic or Newtonian, versus if they're reductionist, like Leibniz, Hume, or Einstein (how they are treated right now in modern physics!) So there's that.

Proceeding from here, perhaps we can characterize math as a property of material objects inhabiting the neighborhood of epiphenomenon.

While I can tell you that it's unlikely you'll be able to account for all of mathematics this way, I shall inform you as to a view known as structuralism in the philosophy of mathematics, where mathematical ideas are really just structures that are instantiated by various things. So physical systems, given satisfying conditions, can instantiate mathematical structures just as ideas in our heads can. However, there are various forms of structuralism with various committal status, and I'm not sure how you would tame those with a materialist doctrine (not saying they're incompatible at all, rather just making a comment WRT my knowledge on the subject).

Immanuel Kant — Garrett Travers

So, one of the key Kantian doctrines, synthetic apriority, had been largely formulated with the example of Euclidean geometry in mind that Kant used. But unfortunately, Kant's ideas were prior to the awareness that there could be what is non-Euclidean geometry: that is, hyperbolic, spherical and the many other we know from our contemporary perspective thanks to the discoveries of Gauss and Schweikart. So the perceived synthetic status of geometry is deflated to analytic status based on axioms and definition, and indeed the sides of a triangle really didn't have to add up to 180 degrees or whatever other purported synthetic facts we knew apriori. These were not in virtue of innate connection of predicate concepts but definition, reducing Kant's project to analytic apriority.