When our understanding of a thing changes, due to shifts in scientific and technological knowledge, it is not simply a matter of reconfiguring our knowledge of the external causal associations between objects. What also changes is the ‘core’ concept of object as center of properties and attributes.

Sure, in some sense. I've long held that, just as Hegel has institutions (e.g. the justice system, family, state, etc.) objectifying morality for a people, we also have science, educational institutions, technology, and the productive arts/trades, serving to objectify certain aspects of the natural world. Positive and negative charge is objectified for us every-time we change a battery or rewire an outlet.

Terminologically though, I would rather say this is a refinement of our intentions, as opposed to our concepts. This is because otherwise, we would be forced to say that "wetness" or "human" is changing, but it seems to be an important distinction that are intentions are changing (and hopefully becoming more perfect). I did not experience a different water when I went swimming before I came to know that water was H2O, a polar solvent, etc.

Plus, to speak only of (presumably efficient) causal associations leaves out the phenomenological whatness of things, their quiddity. I suppose that, on the conception of reason as primarily/wholly ratio, that's all there is, sets of propositions formed from empirical atoms that get shifted around. But I would tend to say that objects are present and grasped/apprehended in a way that transcends this.

Now, something like water does indeed change in some sense when we come to discover that it is a "polar solvent" or "H2O." Being known is a relation after all, and things are, in some sense, defined by their relations. Yet at the same time, there is a more obvious sense in which water today is the same water the dinosaurs swam in, and wetness today the same wetness experienced by a medieval peasant every time it rained. I am not really sure how to capture this distinction outside of an appeal to per se predication.

The reason that this core concept of objectness does mot remain stable in the face of changes in under is that it is an abstraction derived from a system of relations not only between us and the world we interact with, but between one part of the world and another.

I am not sure I really understood this. It seems to me that anything involving "us and the world" necessarily involves "one part of the world and another," so I am not sure what the difference is supposed to be. Nor do I think I wholly understand what changes in the core concept of an object, or abstractions being derived from a system of relations entails.

Not deduced. it is induced by what is discovered in what is observed.

So, you say mathematical proofs are empirical? I think you're on shaky ground there but I now know better than to argue with you about such things, so I'll leave it there.

I'm not saying mathematical proofs are empirical. Divisibility, and thus oddness, evenness and the primeness of small numbers can be directly observed in the ways groups of things can and cannot be divided up. From there abstract rules are formulated. Remember, calculations used to be done on an abacus.

Your reply deserves much more attention.

You offered as a way forward for the different approaches adopted by @Wayfarer and I. offers the Aristotelian account as paradigmatic, which we might come back to later.

You phrased the question as about what the three parsings of "is" in a first order language are able to account for in terms of being and existence, and whether there is more to being and existence than these can be grounded in predication, equivalence and quantification. The issue is now "Will it ever be helpful to use the words 'being' and 'existing' to talk about this ground?" but where the issue is one of terminology rather than concept. I have two issues with this. Firstly that we can't long maintain a distinction between concept and terminology, and secondly that our words and actions work directly in the world and not on our model of the world.



It is difficult to maintain a distinction between what is conceptual and what is terminological, between the structure we accept of how things are and the labels we apply to that structure. This because using a term just is using a concept. This follows immediately from not accepting that there is some thing we can call the "meaning" of the term that is distinct from it's use, but instead looking just to their use.

Two caveats here. Firstly I put this in terms of use, as per Wittgenstein, but it can equally well be put in terms of truth functionality, so as to more closely approach Davidson.

And secondly, the use of "terminology" may mislead folk into supposing that that this view takes concepts to be only linguistic, that it is "language all the way down". That's not what is proposed; rather the concept is the doing.That a cat has a concept of "food bowl" is shown by the behaviour it exhibits, by what it does, by how it uses the food bowl. In this regard, language is just more doing, more behaviour. But - and it is an important but - once language comes into play, there is no going back. The rational structures developed on language cannot be rescinded. Concepts are displayed in our actions, including those actions that involve language. Indeed, our concepts are our actions.



Davidson has shown how trying to explain our behaviour, especially our linguistic behaviour, in terms of conceptual schema leads to irredeemable difficulties. Specifically, that we understand what someone else is doing and why implies that their supposedly different conceptual schema is subsumed by our own. We could not recognise their behaviour as consistent and coherent without thereby making sense of their supposed conceptual scheme. But what this shows is not a similarity in conceptual schemes but that they and we have the same beliefs and act within the very same world. It's not the conceptual schemes that are similar but the world in which we are embedded. Our actions, including our language, are in direct contact with that world.



This post compresses two very large ideas into a very few words. But it might give someone with the right background in Wittgenstein and Davidson an idea of the direction in which this conversation might head. Others will misunderstand. I can't help that.

Oh man, forgive me for saying this, but in the last page of this Thread, we really made a hard turn to Wonderland, didn't we?

I mean, for a Thread called "Mathematical platonism", this just went to shite.

EDIT: And it appears that we haven't gotten out of that Rabbit Hole yet, on page 17 of this thing.

Are you saying divisibility cannot be "divided up" and/or sets displaying "evenness" cannot be divided up? For example, the set of even numbers can be divided up into those even numbers having exactly two 2s.

Humans seem to have evolved to the point of both constructing and exploring mathematics. The counting numbers arise from observations and abilities to distinguish. In my opinion none of math exists in some Platonic realm independent of human brains. These are ideas, not physical objects.

Modern math is concerned more with overseeing the multitude of mathematical ideas and discovering how they relate to one another, than the classical approach to conjuring up problems to solve in the individual areas of the subject (there are about 30,000 Wikipedia pages on math topics, e.g.)

On the other hand, I can't say these ongoing philosophical arguments are of less importance than much of the math being produced. My own areas of exploration are "pure" mathematics and have little to any connections to physical realities. Nevertheless, the results are documented in simple exercises of logic on a set of symbols that are well defined. I don't get the impression that is the case in Platonics. But I could be wrong.

Right, I wasn't asking the second question. I don't think in terms of superior ways of existence—I am not a fan of hierarchical notions of being. — Janus

I realize that, sorry if I implied otherwise. I was just using your question to compare with a type of question that I think others have been asking.

The irony in all this is that I sort of am a fan of hierarchical notions of "being," if by hierarchy we just mean structure or grounding. My idea, not to belabor it to death, is that we'll do a better job by dropping the word "being" to the extent that we can.

the primness of small numbers — Janus

I'm sorry, I can't resist a good typo. Yes, I too find small numbers to be prim, even reticent. But then there's π, which is small but goes on and on forever . . .

The reason that this core concept of objectness does mot remain stable in the face of changes in under is that it is an abstraction derived from a system of relations not only between us and the world we interact with, but between one part of the world and another. — Joshs

@Joshs Hi, I'm not sure if I understood this part correctly. I don't know if you're saying what it seems to me that you're effectively saying there. Can I ask for some clarification there? Specifically: what do you mean, and what do you intend, when you say something like that? Does that question make sense? Let's start with that, I think that could put this Thread back on track.

I'm somewhat familiar with Nozick's politics, which have not inspired me to read his wider philosophical work.

Speaking from this ignorance, if we are going to take philosophical pluralism seriously, shouldn't we avoid the sort of over-arching story found in Philosophical Explanations? Shouldn't we avoid saying that philosophical explanations are thus-and-so?


But to this:

The sense of the world must lie outside the world. In the world everything is as it is and happens as it does happen

Again, this is from the Tractatus, which I take PI to supersede. Roughly, post-PI the "sense of the world" remains unstated, but can be either enacted and shown, or left in silence. In neither case is the sense of the world said.

So this comes down to what we might mean by "expressible" in "is the sense of the world expressible".

It is difficult to maintain a distinction between what is conceptual and what is terminological, between the structure we accept of how things are and the labels we apply to that structure. This because using a term just is using a concept. — Banno

Yes, everything you say is a nice concise view of the problematic territory here. I'm more comfortable with Davidson than Witt on this topic but that's just me.

As I wrote earlier, I need to rethink what I want to say in a way that would be a reply to Davidson, which ain't easy. Maybe the place to start is "Using a term just is using a concept". What if we reply, "Yes, but is using a concept just using a term?" So the question is still, "How, and to what extent, can we dissolve that metaphysical Superglue that seems to bind term to concept?" but reverses the grounding. The Davidson/Witt position would, I think, be that there can't be any grounding because "concept" is parasitic on our terms.

Now you may want to say, "It's not metaphysical Superglue at all, it's the opposite of what metaphysics proposes" and/or "If there is no conceptual scheme, no appeal to shared meanings, but merely a congruence of beliefs, acts, and worlds . . . then what's left for 'concept' to be about?" Those would be meta-challenges, for sure. I need to think more about how I in fact use concepts, and find a couple of paradigm cases of terminological changes that really do hold a concept steady. Then I might be in a better position to restate my case. Should take about a year . . . :smile:

Wayfarer offers the Aristotelian account as paradigmatic, which we might come back to later. — Banno

You and others might find this essay interesting Aristotle was Right After All, James Franklin. (I don't agree with his depiction of the 'other world' of Platonic forms, but it is still pretty much on topic for this thread.)

Are you saying divisibility cannot be "divided up" and/or sets displaying "evenness" cannot be divided up? For example, the set of even numbers can be divided up into those even numbers having exactly two 2s. — jgill

I'm not sure what led you to think I was saying anything like that. It seems to me that arithmetic has its genesis in playing around with groups of actual things and inducing the basic concepts of adding dividing, subtracting and multiplying. Once we have generalized and abstracted those notions and represented them symbolically and formulated the rules that govern them, then all of the elaboraqtions of mathematics become possible.,

In my opinion none of math exists in some Platonic realm independent of human brains. These are ideas, not physical objects. — jgill

Totally agree.

The irony in all this is that I sort of am a fan of hierarchical notions of "being," if by hierarchy we just mean structure or grounding. My idea, not to belabor it to death, is that we'll do a better job by dropping the word "being" to the extent that we can. — J

I also am fine with the notion of hierarchy in the sense of structure or grounding—I just reject the spiritualist notion of degrees of being—you know, the idea that humans are at a higher degree of being than animals, and angels at a higher degree of being than humans, and so on. The "great chain of being" idea I reject.

For me the word 'being' just means, if taken as a noun 'existent' or 'existence' and if taken as a verb, 'existing'.

I'm sorry, I can't resist a good typo. Yes, I too find small numbers to be prim, even reticent. But then there's π, which is small but goes on and on forever . — J

:lol: Nice!

:up: I like the article, since it is saying just what I have been. It's the middle ground between Platonism and nominalism.

.

Nozick's politics — Banno

Yeah, I know, unfortunate. But he's a good meta-philosopher for all that.

if we are going to take philosophical pluralism seriously, shouldn't we avoid the sort of over-arching story found in Philosophical Explanations? Shouldn't we avoid saying that philosophical explanations are thus-and-so? — Banno

Pretty sure Nozick would agree with that. The tone of the book is discursive and investigative, not didactic. It contains one of my favorite passages about doing philosophy:

Philosophical argument, trying to get someone to believe something whether he wants to believe it or not, is not, I have held, a nice way to behave toward someone; also, it does not fit the original motivation for studying or entering philosophy. That motivation is puzzlement, curiosity, a desire to understand, not a desire to produce uniformity of belief. Most people do not want to become thought-police. — Nozick, 13

Roughly, post-PI the "sense of the world" remains unstated, but can be either enacted and shown, or left in silence. In neither case is the sense of the world said. — Banno

True, but evidently it can be referred to. That may be all we need.

Philosophical argument, trying to get someone to believe something whether he wants to believe it or not, is not, I have held, a nice way to behave toward someone — Nozick, 13

But why should that stop us? :wink:

Maybe the place to start is "Using a term just is using a concept". What if we reply, "Yes, but is using a concept just using a term?" — J

Here I'll reach to my other pet philosopher, Austin.

The concept "seven" just is being able to buy seven apples, adding three and four, taking nine from sixteen. There is not a something in addition to these that is the concept of seven. So yes, using a term just is using a concept, but we can do stuff without terms, so using a concept is not just using a term. But better than any of these, just drop the use of "concept" altogether. Drop the concept seven and just add three and four.

The implicit picture is of a "concept of seven" in someone's head that are called when one does things with seven. But if we can add three and four, what further explanatory work is done by the concept?That's why this is muddled:

Brain; ( number system 1 )
Brain; ( number system 2 )
Brain; ( number system 3 ) — Mark Nyquist

But I'm not sure if Mark is advocating or laughing at the suggestion.

...evidently it can be referred to — J

Waved at, perhaps.

I like the article, since it is saying just what I have been. It's the middle ground between Platonism and nominalism — Janus

Indeed. But also note

Aristotelian realism stands in a difficult relationship with naturalism, the project of showing that all of the world and human knowledge can be explained in terms of physics, biology and neuroscience. If mathematical properties are realised in the physical world and capable of being perceived, then mathematics can seem no more inexplicable than colour perception, which surely can be explained in naturalist terms. On the other hand, Aristotelians agree with Platonists that the mathematical grasp of necessities is mysterious. What is necessary is true in all possible worlds, but how can perception see into other possible worlds? The scholastics, the Aristotelian Catholic philosophers of the Middle Ages, were so impressed with the mind’s grasp of necessary truths as to conclude that the intellect was immaterial and immortal. If today’s naturalists do not wish to agree with that, there is a challenge for them. ‘Don’t tell me, show me’: build an artificial intelligence system that imitates genuine mathematical insight. There seem to be no promising plans on the drawing board.

The concept "seven" just is being able to buy seven apples, adding three and four, taking nine from sixteen. There is not a something in addition to these that is the concept of seven. — Banno

As I understand platonism, neither would it. This would be a reification, objectification of the act of act of counting. But it doesn't vitiate the fact that the number is independent of any particular mind, but can only be grasped by a mind.

In the Metaphysics, Aristotle identifies multiple senses of "being," which include:
* Substance (ousia): The primary sense of being, referring to what a thing fundamentally is.
* Qualitative Attributes: Being in the sense of having certain properties (e.g., "the apple is red").
* Existence: Being in the sense of "being there" or existing in time and space (e.g. "the apple is on the table")
* Potentiality and Actuality: Being as a dynamic process, involving what something can become versus what it is. — Wayfarer

The last three can be parsed as predications.

The first is ambiguous, partly about placing something in the domain of discourse and hence making it subject to quantification, and partly about essences, which are more trouble than they are worth.

But it doesn't vitiate the fact that the number is independent of any particular mind, but can only be grasped by a mind. — Wayfarer

I don't get that. What is it for a mind to grasp a number, apart from being able to count to it, add it, or halve it?

Numbers are not things in the head, not mental furniture.

Just looking at what I wrote, maybe multiple forms of mental content are nessecary to deal with numbers.
The number 7 could be a pure number, or a unit of something, like a length, area or volume on and on.
So numbers as mental content are dynamic and modified as needed by mental process.

That exists by observation so why does some secondary Platonic form need to exist?
One form isn't enough?

The "All in one's head" model is a thing known to physically exist. — Mark Nyquist

I'm afraid this does not do the work you need it to do, nor can you bat the ball back to my court so easily. How do you respond to my reasons that numbers can't just be "all in one's head"?

And numbers just as non-physical abstractions doesn't have an explanation.
Give it a try if that's your position. — Mark Nyquist

Think of chess. This is an arbitrary game, with arbitrary rules that exist in our collective heads. It is well known in chess that a bishop is worth 3x a pawn and 1/3 a queen. Impressively, this was known well before computers made it conclusive. Yet, you will never find this in the rules of chess, it was never in anyone's head before it was discovered. How can this be? I think of the rules of chess as creating a "logical landscape", and facts can be discovered in such a landscape that were never in anyone's head. This, despite the rules of chess being 100% arbitrary, having no connection to the actual universe.

Numbers are such a thing. They also have rules which create a logical landscape, about which things can be discovered which were never in anyone's head. But unlike chess, the rules of numbers are intimately tied to the way things actually work in the universe. If you have one unit, and combine it with another unit, you get two units, no matter how you define what a unit consists of. So long as the definition of unit is consistent, this works for anything across space and time.

I don't think this is an answer in itself, and I'm no mathematician. I'm just trying to convey my intuition on how the problem can be thought about, without resorting to "all in the head", and without resorting to mystical Platonic essences...

Think of chess. This is an arbitrary game, with arbitrary rules that exist in our collective heads. It is well known in chess that a bishop is worth 3x a pawn and 1/3 a queen. Impressively, this was known well before computers made it conclusive. Yet, you will never find this in the rules of chess, it was never in anyone's head before it was discovered. How can this be? I think of the rules of chess as creating a "logical landscape", and facts can be discovered in such a landscape that were never in anyone's head. This, despite the rules of chess being 100% arbitrary, having no connection to the actual universe. — hypericin

This is an excellent point. I would even add: when someone plays a game of chess, it would be wrong to think that the chess-player is executing monarchic politics when he moves the piece called "the King". I mean, come on, have you ever thought such a demented thing while playing an actual game of chess? Of course not. But you see, this is what I'm humbly saying: philosophers step in at this point of the dialogue, and they say: "How do you know that you're not really executing monarchic politics when you're playing a real game of chess at the park with a bunch of random people?" And the appropriate response to that sort of question, is a Moorean response.

Such is my sentiment on that issue.

I don't get that. — Banno

We’re talking about the faculty of reason. I think we take it for granted without noticing how significant it is. Consider what it enables.

It might just be that our mental process is so effortless that we ignore it.

So existing or not existing is one question.
Platonism as an abstraction doesn't exist.
Mental content does physically exist.

Rules that exist in mental content can be valid or invalid. But as mental content they both physically exist.

It's

...only be grasped by a mind — Wayfarer

that is problematic. Again, what might it be for a mind to grasp a number, apart from being able to count to it, add it, or halve it?

If you can count out seven things, do additions that result in or use seven, double and halve seven... what more is there that you are missing, that is needed before you can be said to have grasped seven?

I don't think there is anything more to grasping seven than being able to use it. Hence concepts are no more than being able to work with whatever is in question, and thinking of them as mental items in one's head is fraught with complications.

Friends, if I may:

There is evidently an elephant in the room, and no one is clearly addressing it. Perhaps because you find it somehow awkward (I don't actually know what that would take in a Thread called "Mathematical platonism", but I digress).

The "elephant in the room" here, so to speak, presents itself as "pack". A "multiple", if you will. A nice, shiny-looking Pandora's Box of philosophical problems.

So let's be simple about this, in a methodological sense. Let's simply, as engineers would. Let us address one specific "part" of that so-called "Elephant in the Room":

Geopolitics, but from the "point of view", so to speak (i.e., the "conceptual framework") of Game Theory.

This is what is now under discussion in this Thread, it seems. In some way or another. Perhaps you had a simpler picture in mind. Perhaps it was instead more complicated than mine. But that is more or less the "tone", if you will, that is being "played" (as in, "Playing the Classical Harp") here. As in, "Oh, Aristotelian Realism, how Romantic, it is so Lovely, and Yet So Troubled".

And what I'm saying, "mates", is that the solution is very, very simple:

Australian Realism (is greater than, in the mathematical sense) Aristotelian Realism.

So, let us simplify that, shall we?

Australian Realism -> Aristotelian Realism.

Which one of you would like to throw yourselves over that metaphorical grenade of a thesis? I'll tell you one thing: it certainly won't be me.

(I, Arcane Sandwich, have edited this thread for Clarity's Sake. And here's the joke: who is "Clarity"? What is her sake in all this?)

Pretty much right.

One can account for this by understanding commissive speech acts. These are speech acts that bring something about. An example would be "I name this ship the King Charles", performed by the designated dignitary at the proper time and place - before that act, the ship has another, or even no, name. After, and in virtue of, that act, the ship comes to be named the "King Charles".

The act counts as naming the ship.

The bishop on a chess board can be moved in any direction, along a row, along a diagonal, picked up and put in the place of the opposing queen, and so on. But only a move along a diagonal counts as a move in a game of chess.

That "logical landscape" of which you speak is constructed using this sort of structure - taking something and making it count as something new. So yes, "If you have one unit, and combine it with another unit, you get two units, no matter how you define what a unit consists of" because that now counts as two units.

So yes, "If you have one unit, and combine it with another unit, you get two units, no matter how you define what a unit consists of" because that now counts as two units. — Banno

Yes, IF you have one unit. It does not follow that you do have one unit to begin with.

Think of it like the three positions in the Analytic Metaphysics of Ordinary Objects.

There, in that context (the metaphysics of ordinary objects), there are three (and only three) logical answers to van Inwagen's Special Composition Question, or SCQ for short:

1) Never (nihilism)
2) Sometimes (particularism)
3) Always (universalism)

Why is this important? Because it is itself undercut by (or perhaps is parallel to) a similar distinction, this one entirely metaphysical, not mereological like the previous one:

1) Eliminativism: there are no ordinary objects, nor extraordinary objects.
2) Conservatism: there are only ordinary objects, and there are no extraordinary objects.
3) Permissivism: there are both ordinary objects as well as extraordinary objects.

From a purely Pragmatic Point of View, the second option is the most practical one in cost-efficient terms. It "gets the work done", which is something that Eliminativism cannot even do, while it avoids incurring in enormous metaphysical (and by extension, mereological) costs. It is simply the greatest solution in terms of metaphysical cost-effectiveness.

Is this too "rambly", for the OP of this Thread?

P.S.: Can someone just explain the dumb joke about "Edited for the sake of clarity" versus "Who is Clarity?" If you explain it correctly, I'll award you with the fictional "Immaterial Medal of Greatness".

that is problematic. Again, what might it be for a mind to grasp a number, apart from being able to count to it, add it, or halve it? — Banno

Numbers have significance apart from counting, for instance there are four gospels in the Bible because there were four elements. Four is a symbol of the earth because there are four directions. Most people in my world know what 666 means, and so in.

This doesn't diminish your point, that numbers are used, just that counting isn't all they're used for.

Sure.