These two responses show a similar approach to the problem, which I think is mistaken. We shouldn’t be conceiving of the “Distinctively Logical Explanation” question as a question about how to define terms, or about whether language can resolve the question. The DLE problem assumes a linguistic stipulation that is uncontroversial or at least agreed-upon, or at the very least accepted for the purposes of discussion. With this stipulation in place, we then go on to ask about the relation between logic-or-math-as-necessity and the alleged necessity of events in the world, such as the 23 objects that can’t be evenly divided, or the cat that can’t be here and in Paris too.

It is trivially true than any given definition of an “object” will determine what we can say about it, using math or anything else. The larger puzzle is this: How is it the case that, no matter what definition we use, we discover these regularities between math/logic and the world? Even the bizarre definitions of “cat,” once accepted, have unambiguous consequences in terms of regularities. Wayfarer calls this a case of the world “mysteriously agreeing with our abstractions,” but that begs the question. Is it the world that is doing this, or are our abstractions mysteriously agreeing with the world? Are there abstractions that “agree” better than others? This is the Phillips-head screwdriver problem. We can just accept the agreement as a brute fact, not mysterious at all, or we can claim a coherent evolutionary explanation, or we can continue to ask why. My only point here is that I don’t think we should look to language for a solution. Resolving a linguistic ambiguity won’t tell us whether math/logic is a genuine causal constraint on the world.

are our abstractions mysteriously agreeing with the world? — J

This becomes a ridiculous question as soon as you understand that "abstract" literally means "taken out".

origin of abstract
Middle English: from Latin abstractus, literally ‘drawn away’, past participle of abstrahere, from ab- ‘from’ + trahere ‘draw off’

https://en.bab.la/dictionary/english/abstract

So where are abstractions taken from? I suggest "the world" is a sensible answer, and one that explains the "mystery" rather well.

I can see that you wouldn't like this approach on the grounds that it shoots your fox and spoils the fun of the chase.

So where are abstractions taken from? I suggest "the world" is a sensible answer, and one that explains the "mystery" rather well — unenlightened

It is indeed a sensible answer, but doesn't explain what appears to be the modally necessary character of the abstractions, and their role in explanation, if any. Have you read the target paper I cited? It might explain the problem better than I have.

I can see that you wouldn't like this approach on the grounds that it shoots your fox and spoils the fun of the chase. — unenlightened

My fox is quite healthy still, thank you! And anyway, I'm an animal rights activist and must urge you not to shoot at any foxes, real or metaphorical. :wink: (You can imagine how I grit my teeth whenever Ted Sider (and Plato) go on about "carving reality at the joints.")

It is the "I" that sees a relation between many different objects in the world. It is not the world that is relating a particular set of objects together. — RussellA

And yet the world presents to us regularities that we capture in empirical research. The regularities that our minds create and the regularities of nature is a tricky subject. Kant, for example, seemed to conflate the two as part of the same "transcendental" constraints that our minds impose on "the thing-itself". Yet, this seems to be at odds with our usual intuition that something empirical, is in some sense a part of "the world itself', not just our minds' way of translating the world. We aren't translating perhaps, but simply copying what is the case- the usual "idealist vs. realist" debate. So the math works because "the patterns are real", or the math works because our minds think in terms of these regularities when it imposes itself onto the universe. Well, certainly, our language-based minds create "objects" from the anarchy of the environmental input. Yet, when our minds impose such things, it also sees that there are various regularities that constantly present themselves that are NOT just patterns, concept-creation, and syntactic manipulation that our brain creates ("notepad".. "thing".. "blob".. "amorphous shape".. "weird unknown object" "car", etc. etc.). Gravity, electromagnetism, chemical interactions, biological interactions, etc. work ways that impose on us their workings, not the other way around.

It is indeed a sensible answer, but doesn't explain what appears to be the modally necessary character of the abstractions, and their role in explanation, if any. Have you read the target paper I cited? It might explain the problem better than I have. — J

Yes. I note that causation is also an abstraction, and that there is not 'necessarily' more than one object, and leave you in peace.

I see what you mean, but we can construct an infinite number of worlds with different abstract entities highlighted (see "grue and bleen", Sider, p. 16) and most of them won't "work" at all, if by "work" you mean "give us a useful conceptual basis for navigating the world." — J

Correct.

Yet there is nothing wrong, logically, with the way these abstractions are being matched to reality. — J

If reality is not contradicting those identities, then they hold. Meaning we can identify reality in multiple ways as long as reality does not contradict our claim. The moment reality does contradict our claim however, its over. For example, if I view that every time I touch a statue, it rains, the time when I touch a statue and it doesn't rain, my abstraction is contradicted and needs to be amended or discarded to continue to be a logical match with reality.

Yes, that all seems right to me. The question you were raising, though, was about particular "matches with reality" that, in addition to being logically consistent and uncontradicted by the facts (such as grue and bleen), also constitute an "identity" -- perhaps like natural kinds? I just think this needs further explanation; logic and noncontradiction alone won't get us to why some matches seem more natural or reality-mirroring than others ("privileged structure").

The larger puzzle is this: How is it the case that, no matter what definition we use, we discover these regularities between math/logic and the world? — J

There are some questions that are problems of language, such as Q2 and Q4. Q2 is dependent upon the definition of "object" and Q4 is dependent on the definition of "cat".

Q1 is also a problem of language, in that dividing 23 objects by 3 gives $7\frac{2}{3}$ things. But Q2 defines a fraction of an object as not being an object, meaning that by definition the number 23 is not divisible (evenly) by 3.

I agree that there are, however, some questions that are not problems of language, such as the equation
${d = 0.5 * g * t^{2}}$ which accurately predicts where a dropped object will be at a given time.

I agree that the solution as to why there is such a good agreement between the equation ${d = 0.5 * g * t^{2}}$ and what we observe in the world is not in language, in that any definition of "object" is irrelevant.

As regards Q3 and the LNC, the propositions "p is the case" and "p is not the case" are mutually exclusive. But in fact it may be difficult to find an example of "p" that can actually be used. For example, as regards problems of language, "half an apple is an object is the case" according to John but "half an apple is an object is not the case" according to Mary. As regards problems not of language, "${d = 0.5 * g * t^{2}}$ is the case" as far as we know but "${d = 0.5 * g * t^{2}}$ is not the case" may be true. The ambiguities in thought are such that an clear-cut example of the LNC may be difficult to find.

To my understanding, we invent an equation and check whether it conforms to what we observe in the world. If it doesn't then we discard it, and if it does then we keep it. We keep the equations that work. In fact we don't need to know why a particular equation works as long as it does work.

However, the fact that an equation such as ${d = 0.5 * g * t^{2}}$ has consistently been shown to work over a long period of time is no guarantee that it will always work, in that any agreement between the equation and what is observed in the world may be accidental, as pointed out by Hume's concept of the constant conjunction of events.

In answer to your question, if we have invented a maths/logic founded on structural regularities, and discover regularities between our maths/logic and the world, this infers that the world is also founded on a structural regularities.

The DLE problem assumes a linguistic stipulation that is uncontroversial or at least agreed-upon, or at the very least accepted for the purposes of discussion. With this stipulation in place, we then go on to ask about the relation between logic-or-math-as-necessity and the alleged necessity of events in the world, such as the 23 objects that can’t be evenly divided, or the cat that can’t be here and in Paris too. — J

I understand the point you're making, but I want to push back on this a little.

The most interesting thing going on here is the creative leap of enlarging your mathematics to include fractions as well as integers.

And their use in solving problems of sharing is probably the most common way fractions are introduced to children. Banning them "for the sake of argument" is depressing, and gives the whole problem an air of artificiality I suspect it deserves.

Sorry, @J, I would like to be more enthusiastic ― I've even made a first pass at the paper ― but surely this is all just beating the analytic/synthetic horse to death yet again. Compare that to the invention of fractions, and which is more impressive?

A different world, if there could be such, would reveal different regularities, but the role of math would be unchanged. — J

I think the interesting question is, then, whether 'our' mathematics would be 'true in all possible worlds'. Meaning, perhaps, that it's not really 'ours'!

So where are abstractions taken from? I suggest "the world" is a sensible answer, and one that explains the "mystery" rather well. — unenlightened

But you said:

If you have 23 objects you have already mathematicised them by counting — unenlightened

Apparently you should have said, "If you have mathematized objects you have already had recourse to the 'pre-mathematical' world."

If the abstraction of mathematics is derived from the world, then the indivisibility of the 23 is more than a merely mathematical fact.

Are mathematical truths necessary if mathematics is grounded in the contingency of the world?

Gravity, electromagnetism, chemical interactions, biological interactions, etc. work ways that impose on us their workings, not the other way around........................Kant, for example, seemed to conflate the two as part of the same "transcendental" constraints that our minds impose on "the thing-itself". — schopenhauer1

Does the world impose itself on the mind or does the mind impose itself on the world?

Perhaps its a combination of both.

We observe regularities in the world.

We then invent the equation ${d = 1.0 * g * t^{2}}$, discover that it doesn't work, and discard it. We then invent the equation ${d = 2.0 * g * t^{2}}$, again discover that it doesn't work, and discard it. Eventually, after many attempts, we invent the equation ${d = 0.5 * g * t^{2}}$, discover that it works, and keep it. In one sense, the world has imposed itself on us, in that the world has "determined" that the equation ${d = 0.5 * g * t^{2}}$ works, not us.

However, in another sense, we impose the equation ${d = 0.5 * g * t^{2}}$ onto the world, in that following Hume's concept of knowledge by constant conjunction, any correspondence between the equation and the world may be accidental. Today the equation may work, but tomorrow it may not. We only know in a pragmatic sense that the equation does work. We don't know why it works. Because we don't know why the equation works, we are not able to say that it will always work, as the equation doesn't contain within itself its own proof.

The world imposes itself on us which equation we use, but we impose our equation onto the world, even though the equation may not correspond with any underlying reality within the world.

Are mathematical truths necessary truths
Following the schema "snow is white" is true IFF snow is white as a definition of "truth", then "
${d = 0.5 * g * t^{2}}$" is true IFF ${d = 0.5 * g * t^{2}}$

But the mathematical equation "${d = 0.5 * g * t^{2}}$" has originated from observations of constant conjunctions within the world (using Hume's term), it hasn't originated from a proof derived from a knowledge of the fundamental reality of the world.

Therefore, we don't know if it is the case that in the world ${d = 0.5 * g * t^{2}}$. It then follows that we don't know if "${d = 0.5 * g * t^{2}}$" is true. We know it works, but we don't know if it is a necessary truth.

Kant and a Transcendental Deduction that mathematical truths are necessary truths

In B276 of the CPR, Kant uses a Transcendental Deduction to prove the existence of objects in the world.

As the equation "${d = 0.5 * g * t^{2}}$" does successfully and consistently predict what is observed in the world, we could use a similar Transcendental Deduction to prove that in the world is the underlying reality that ${d = 0.5 * g * t^{2}}$.

Using such a Transcendental Deduction, we could unify a world that imposes itself on the mind and a mind that imposes itself on the world.

I just think this needs further explanation; logic and noncontradiction alone won't get us to why some matches seem more natural or reality-mirroring than others ("privileged structure"). — J

You may be interested in reading this then. https://thephilosophyforum.com/discussion/14044/knowledge-and-induction-within-your-self-context/p1

There's a fantastic summary the next post after mine. If you're serious about this, I would read it.

I see what you're saying. Let me try to work up an example that is less controversially stipulative. As for the analytic/synthetic distinction, I'm not sure we really have to go there. Perhaps it won't do as a description of the difference between logic and facts-about-the-world, but wouldn't you agree that Jha et al. are pointing to something that can be talked about, and represents a genuine question? Or perhaps you wouldn't.

I did read the summary. Is this the passage you're referring to (concerning "privileged structure" or the like)?:

"Applicable knowledge is when a form I have created in my mind, fits reality. The simpler the properties in the distinctive knowledge, the simpler the applicable knowledge accrued. If the essential properties of sheep are curly fur and hooves, this would be indistinguishabl[e] from for example a goat."

↪Philosophim I did read the summary. Is this the passage you're referring to (concerning "privileged structure" or the like)?: — J

This is part of it, yes. "Privileged" knowledge is really just simple knowledge that has been tested and confirmed so tightly as to be assumed to be 'true'. 'True' in this case being beyond all doubt or viable questioning at this point in time. The creation of our identities has been refined to match reality in ways that are currently impossible to contradict, and are so fundamental and basic as to not rely on much else for their foundation.

It is from these that we generally build other 'less stable' knowledge. If you got to the part about induction, you'll realize that the deduction required for knowledge is highly expensive in time, effort, and perception. Sometimes we reach a point in trying to acquire knowledge that we reach limits that must be filled in with induction. The hierarchy of whether an induction is more cogent than another is probability, possibility, plausibility, and irrational. When comparing inductions, if there is an induction that is at a higher tier, it is more rational to choose that over the lower tier.

For example, the probability of winning a lottery is 1 in 10 million. It is possible to win the lottery. What induction is more rational to consider if you are deciding to spend money on a ticket? The first one. Its possible to win the lottery, but highly unlikely. Now imagine a lottery that costs a dollar per ticket that has a 1 in 2 chance of winning millions of dollars. Same thing. Its highly likely we will win it versus the cost to entering. Compared to this, the idea that 'Its possible to win the lottery" is an inferior induction to reason with.

If you think of knowledge as often complex structures that are built upon other knowledge, more complex structures of knowledge often rely on induction of some kind here and there. The more 'solid' the knowledge, the less it relies both on inductions, but lower tiers of induction. Fundamental bits of knowledge like math are relatively uncomplex, built on the basic structure of 'the logic of distinctive experiences'. Because there is little to no induction involved, or the induction that we do rely on is the best option that we have, we consider these 'privileged'.

I realize I've been using the term "privileged structure" without much explanation, as if everyone is familiar with Theodore Sider. Let me expand a little. Sider uses the "grue and bleen world" example (which you can read about here, p. 16) to refer to a situation that he believes needs explaining: If we encountered a people who used grue and bleen as their concepts, we'd be unable to fault them on any logical grounds. Nor would anything in the world contradict their choice of these concepts, as concepts -- there really are grue and bleen things, and we know how to say true and false things about them. They would of course prove completely unworkable in practice. But why? What makes our green-and-blue conceptual world better? Is it simply a pragmatic question? But that only pushes the question back a step, for now we have to ask, Why does it work better? Is that just lucky for us, or is there something about the world, and its structure, that green-and-blue reflects with more than mere accuracy? If so, that would be a privileged structure in Sider's sense, and the grue-and-bleen world would not.

So, on that understanding, how would you explain privileged structure? I can see how you can demonstrate the pragmatic success, but what's the next, explanatory step?

Sider uses the "grue and bleen world" example (which you can read about here, p. 16) to refer to a situation that he believes needs explaining: If we encountered a people who used grue and bleen as their concepts, we'd be unable to fault them on any logical grounds. — J

Correct. We would not be able to fault them. Name creation is simply that, name creation. If you read my paper, I actually cover this with a sheep and a goat a bit. There can be a person, and thus a society, that calls both a sheep and a goat, a goat. This is because in their eyes, the essential properties of the sheep and goat, "Fur and hooves" are all that matter. The fact one has weird horns or eyes is a non-essential proper for them. Its irrelevant.

But such broad definitions may run into problems if one were to start raising 'goats'. You would find that one type of goat has medical issues that the other doesn't. They behave differently when managing them around your pens. These differences start to elevate in importance, so they become more essential. One could decide "These two are so different, I'm going to start calling one a sheep," or "(Referring to sheep) I'm going to start calling these 'fluffy goats'".

So with color, it would be the same. To a color blind person, there is no 'red' for example. In most cases, its irrelevant. However, when someone creates a bit of art with color, or you have a need to identify things based on red coloration, this becomes a problem with accurately making decisions about reality.

In my opinion, there are a few factors that determine a cultural set of words and identities.

1. Real life effectiveness

This is actually the most impactful reason. Identifying things incorrectly often leads to mistakes, stumbling blocks, and inconveniences. This gets a person and/or society to change if there is a better alternative.

2. Fulfills emotional desires

Maybe there is a viable reason to use 'grue', but since it doesn't personally impact my life, and I simply 'don't like it', I'm not going to use it. The phrase "Gay marriage" has nothing logically wrong with it, but for some people it made them uncomfortable, so they avoided it. Its the same reason I don't use "Oh snap!" when I make a mistake. It just feels dirty. :D

3. Fulfills a power structure

Sometimes words and phrases contain a cultural power over people and societies. The term "God" might not be clear or particularly useful other than a means of getting a people to unite as a nation "Under God". "Don't use the term transsexual, that's offensive, use transgender", is another example of using perfectly descriptive words to control a narrative.

Kant and a Transcendental Deduction that mathematical truths are necessary truths

In B276 of the CPR, Kant uses a Transcendental Deduction to prove the existence of objects in the world.

As the equation "d=0.5∗g∗t2

=
0.5
∗

∗

2
" does successfully and consistently predict what is observed in the world, we could use a similar Transcendental Deduction to prove that in the world is the underlying reality that d=0.5∗g∗t2

=
0.5
∗

∗

2
.

Using such a Transcendental Deduction, we could unify a world that imposes itself on the mind and a mind that imposes itself on the world. — RussellA

Good stuff, but the question becomes, "Are the equations being imposed or simply reflected in the mathematics?". Some neo-Logos philosophies might say the mind cannot but help seeing the very patterns that shape itself. However, it need not be so congruent.

I can imagine a type of pattern whereby the mind works (X), and a pattern whereby the world works Y, and X may be caused by Y, but X is not the same as Y. They may be contingently related, but one happens to "loosely" understand the other rather than necessarily understand the other. Does this distinction I am describing make sense? And then, if you get what I am saying, how do we make sense of it? Which is it? Is our language contingently relating with the world or necessarily relating to the world.

I can see a sort of holistic beauty in the aesthetic of the language reflecting the world because it is derived from (the patterns) of the world. The beauty of the golden ratio, the spiral, a pattern, a smooth surface, a continuation, etc.

However, I can see a sort of nihilistic "contingency" in the aesthetic of language never really derived from, but only loosely reflecting the world. There is a disconnect between the logics. This is the horror and anxiety of remoteness, disconnect, discrete, contingency.

As for the analytic/synthetic distinction, I'm not sure we really have to go there. Perhaps it won't do as a description of the difference between logic and facts-about-the-world, but wouldn't you agree that Jha et al. are pointing to something that can be talked about, and represents a genuine question? — J

Doesn't this remind you of "Two Dogmas of Empiricism"? Quine's target was the atomic proposition, and he intended to show that there's no hope of teasing out which parts of our beliefs were analytic and which synthetic.

Here we have explanations. Which part of the explanation is pure math and which part involves facts about the world?

Sound familiar? Even the form of the argument struck me as similar but I haven't made a side-by-side comparison.

The result, as before, is that you cannot tease out any supposedly pure math part, roughly.

So I assume, in the great sweep of things, Lange is fighting Quine, and Jha defending Quine, roughly.

It's understandable. Quine himself had very mixed feelings about whether the laws of logic were subject to revision. I think his final answer was yes, but it's a last resort, and they are very insulated, resistant to revision.

I could be way off here, but that was my impression as I was trying to decide whether to really dig into the paper.

The article made me think about this passage from an interview with physicist Sabine Hossenfelder:

Q. You claim that a person’s information, if we trust mathematics, is still there after death, dispersed throughout the universe, forever. Are we immortal?

A. If you trust the mathematics, yes. But it is not an immortality in the sense that after death you will wake up sitting in hell or heaven, both of which – let’s be honest – are very earthly ideas. It is more that, since the information about you cannot be destroyed, it is in principle possible that a higher being someday, somehow re-assembles you and brings you back to life. And since you would have no memory of the time passing in between – which could be 10¹⁰⁰ billion years! – you would just find yourself in the very far future. — interview with Sabine Hossenfelder

I think we probably do a certain amount of explaining by way of the dictates of math, but much more frequently, we make predictions with math. We assume that if our predictions are wrong, it's not math that failed, but our powers of reasoning.

Why is math so faithful? It may be that we can't know that.

Cool comparison, I hadn’t thought of it!

I don’t think I agree with your interpretation, though. You recall that Quine’s target was meaning-synonymy as a supposed criterion for analyticity. He readily acknowledges, at the start of the paper, that logical truths are excluded from his criticism. So we have to ask, is there a “parallel exclusion” in the case of explanations that include part math, part facts-about-the-world?

Taking the math part to be parallel with “analytic,” we want to know whether maths are logical truths (and thus both easily identifiable and unexceptionable, according to Quine), or whether they are more like meaning-synonymy statements. Frege may be helpful here; he also divided analytic statements into two groups. The first is Quine’s “logical truths”; the second is supposed to be reducible to logical truths on the basis of purely logical definitions. As Susan Haack points out (in her Philosophy of Logics), this would mean that “the truths of arithmetic are, in this sense, analytic.” (And Kant, of course, would disagree.)

Do Jha et al. take a roughly Fregean stance here, concerning the relation of math to analyticity? They don’t address this directly, to be sure, but I think they do. The reason lies in their reasons for rejecting distinctively mathematical explanations (DMEs) in the first place. Math, according to them, can’t play an explanatory role in scientific explanations because it can’t say anything about “the world,” due to its a priori nature. Now I know Kant though math could be both a priori and synthetic, but that has never struck me as plausible, and I think we should go with Frege. (And anyway, as @schopenhauer1 pointed out, the synthetic nature of math for Kant is transcendental. It operates as we structure experience, it’s not something we learn “in the world.”)

So if we attribute the Fregean stance to Jha et al., then they don’t say that “you cannot tease out any supposedly pure math part, roughly”. It’s precisely because you can do this that DMEs won’t work.

Still, it’s not a simple question, and I’m not sure I’m right.

Quine himself had very mixed feelings about whether the laws of logic were subject to revision. I think his final answer was yes, but it's a last resort, and they are very insulated, resistant to revision. — Srap Tasmaner

Just as an aside, I think Quine believed the laws of logic were true because we could supply clear definitions for all the operators and connectives. This is in Word and Object. In a subsequent work which I haven’t read, The Philosophy of Logic, he extends this to non-classical logics, according to Haack. She says that he accepts “a meaning-variance argument to the effect that the theorems of deviant and classical logics are, alike, true in virtue of the meaning of the (deviant or classical) connectives; which, in turn, seems to lead him to compromise his earlier insistence that fallibilism extends even to logic.” So it sounds like your "very insulated, resistant to revision" is spot on.

Why is math so faithful? It may be that we can't know that. — frank

Neoplatonic mathematics is governed by a fundamental distinction which is indeed inherent in Greek science in general, but is here most strongly formulated. According to this distinction, one branch of mathematics participates in the contemplation of that which is in no way subject to change, or to becoming and passing away. This branch contemplates that which is always such as it is and which alone is capable of being known: for that which is known in the act of knowing, being a communicable and teachable possession, must be something that is once and for all fixed. — Jacob Klein, Greek Mathematical Thought and the Origin of Algebra.

It's the inherent mysticism of Platonic realism that analytic philosophy finds distasteful.

(Empiricists) view Platonism with skepticism. Scientists tend to be empiricists; they imagine the universe to be made up of things we can touch and taste and so on; things we can learn about through observation and experiment. The idea of something existing “outside of space and time” makes empiricists nervous: It sounds embarrassingly like the way religious believers talk about God, and God was banished from respectable scientific discourse a long time ago.

Platonism, as mathematician Brian Davies has put it, “has more in common with mystical religions than it does with modern science.” The fear is that if mathematicians give Plato an inch, he’ll take a mile. If the truth of mathematical statements can be confirmed just by thinking about them, then why not ethical problems, or even religious questions? Why bother with empiricism at all? — What is Math?

OK. So no one of your factors would be something like "This set of concepts more accurately reflects the ontological structure of the world"? You'd rely on pragmatic and/or personal-preference explanations for the chosen set of concepts?

Let me try to work up an example that is less controversially stipulative. — J

How about this?

For "object" in the "23 objects" question, let's say "chickens." We wish to divide 23 living chickens evenly among 3 people; we discover this can't be done. Here there's no question of how to define the object, or whether fractions can ride to the rescue. So: Is this division impossible for the same reason -- a distinctively mathematical one -- that the number 23 can't be divided evenly by 3? Or, if you don't accept "distinctively mathematical," how would you characterize it?

Are you sure we should call something like d=0.5∗g∗t2 a mathematical truth? I thought it was only true on some interpretation; as it stands, it has no meaning.

OK. So no one of your factors would be something like "This set of concepts more accurately reflects the ontological structure of the world"? — J

More, "This set of concepts most accurately represents what can be known about the world."

In your other thread we touched on the Scholastic transcendentals or convertibles. Another transcendental besides being and truth is oneness (unum). — Leontiskos

In this vein, this paper looks interesting, "being without one: deleuze and the medievals on transcendental unum."

There was consensus among the scholastics on both the convertibility of being and unity, and on the meaning of this ‘unity’—in all cases, it was taken to mean an entity’s intrinsic indivision or undividedness. [19] In this, the tradition was continuing and affirming a definition first proposed by Aristotle in the Metaphysics. [20] This undividedness, in the words of Aquinas in his Commentary on the Sentences, is said to lie “closest to being.”[21] For the most part, ens and unum were distinguished by these thinkers only logically or conceptually—unum adding nothing real to being, or more properly, adding only negation, only a privation of actual division.[22] It was common practice in medieval philosophy to distinguish the transcendental sense of unum, running through all of the categories, from the mathematical sense of unum, restricted to the category of quantity. These two ‘ones’ are each in their own way opposed to ‘multiplicity.’[23] Aquinas offers a succinct account of this in his Summa Theologiae (Ia. q. 11, art. 2).[24] The ‘one’ of quantity is the principle of number; it is that which, by being repeated, comprises the sum (the multiple).[25] Aquinas says that there is a direct opposition between ‘one’ and ‘many’ arithmetically, because they stand as measure to thing measured, as just-one to many-ones. Likewise, transcendental unity is opposed to multiplicity, but in this case not directly. Its opposition is not to the many-ones per se, but rather to the division essentially presupposed in and formal with respect to the multiplication of actual multiplicity. This tracks with a consistent distinction in Aquinas between division and plurality in which division is seen as ontologically and logically prior.[26] Transcendental unity then, has a certain priority to its predicamental counterpart.

We will return below to the consequences for contemporary ontology that follow upon this fact that, in its developed form, it was division, not plurality, that was taken by the classical tradition to be the precise contrary to transcendental unity. . . — Being without One, by Lucas Carroll, 121-2

Unum in the same sense as in non-dualism, advaita, non divided.