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. — Count Timothy von Icarus

The concept of ‘object’ serves a purpose for us. It allows us to unite features and attributes into a single ‘this’. The concept of feature or attribute also serves a purpose for us. It allows us to anticipate how an aspect of the world will respond to our investigation of it. When we turn our head one way, we anticipate it will move in the other direction. When we walk around it, we expect to see another side of it. So in a sense, objects are instructions for how to anticipate responses to our actions on a part of the world. Knowing about water is know how it will respond to our touching it , moving in it, exposing other objects to it. Knowing about water is also knowing where it came from, how it was created and how it can be transformed. These are important to us when we want to interact with it in special ways.

We could say that the object is an anticipatory dance between us and a part of our surroundings. But notice how the ‘same’ object changes depending on the pattern of this dance. Take an object as simple as a point. Does it makes sense to talk about a point i swore t of the nature of the way we dance with it?

If the composition of points out of lines or of lines out of points is conventional rather than factual, points and lines themselves are no less so. ... If we say that our sample space is a combination of points, or of lines, or of regions, or a combination of combinations of points, or lines, or regions, or a combination of all these together, or is a single lump, then since none is identical with any of the rest, we are giving one among countless alternative conflicting descriptions of what the space is. And so we may regard the disagreements as not about the facts but as due to differences in the conventions-adopted in organizing or describing the space. What, then, is the neutral fact or thing described in these different terms? Neither the space (a) as an undivided whole nor (b) as a combination of everything involved in the several accounts; for (a) and (b) are but two among the various ways of organizing it. But what is it that is so organized? When we strip off as layers of convention all differences among ways of describing it, what is left? The onion is peeled down to its empty core. (Nelson Goodman)

Note that these conventions are not just ways of describing a thing. They are instructions for how iinteract with it in particular ways in order to achieve a predictable series of responses from it. How that object appears to us is a function of what we are doing with it, how we are dancing with it. To ask if the planet Jupiter exists is to ask about a particular sort of interactive dance. It makes no more sense to imagine Jupiter the planet independently of some convention of practical engagement with it than it does to imagine a tango with only one performer, or a duet with only one singer. Is a photon a particle or a wave? It depends on which apparatus is dancing with it.

Again, it's not that someone can play various major chord, record and read them, and recognise them when they hear them, and yet not have, or not understand, what a major chord is, because they are missing something more... the concept. — Banno

OK, that's clearer to me.

All language stops with showing and doing.

But again, I don't think I've quite understood your point. — Banno

My point -- somewhat off the point, perhaps -- is that we never arrive at something we can simply take as what it is, as opposed to "counts as." Or do we? This is reminiscent of that Wang essay about Davidson, a while back.

To make his argument go through, I think Wang has to show not only that common-sense experience is possible, but that the other kind – raw, unmediated perceptions, "thin experience" – is impossible. — J

Should we say that, at some given level of demonstration, we have "raw, unmediated perception"? Something we can point to and say, "This," sans interpretation?

Sure humans evolved, and so too the ability to count, speak, tell stories and much else besides. But that doesn't mean that Frege's 'metaphysical primitives' such as integers and logical principles, can be legitimately depicted as a result of evolution. The aim of evolutionary theory is to explain the origin of species, not an epistemology. — Wayfarer

But there is currently an evolutionary explanation of epistemology underway, and of science more generally. For now it's just a research program in the perhaps Lakatosian sense, but they have not produced any opinion-swaying papers just yet.

That was an excellent thread. Yep, as you said there, it's not that conceptual schemes can't be relative, but that there can't be conceptual schemes.

"This," sans interpretation? — J

And again, PI§201. There's a way of understanding that is not seen in giving an interpretation, but playing the chord - or maybe changing the key of the tune.

There's more we might do with the OP. Might attract @Michael back from his holiday. Might even interest @jgill.

Michael chose infinitesimals with due consideration. There's an issue as to whether they exist or not... odd little things as they are.

ε is an infinitesimal iff 0 < ε < 1/n for all n ∈ ℕ

But this sort of account doesn't tell us what they are, only what they are not — they are not the reciprocal of any natural number. This pisses constructivists off, becasue they like to have an account of what something is before they commit to it's existing. IF there are only natural numbers then there are no infinitesimals.

The account I offered is more permissive than constructivism, since it says that since we can pretend that there are infinitesimals, we might as well say there are infinitesimals. After all, all numbers are just such pretence - "counts as".

This permissiveness might be my downfall.

Should we say that, at some given level of demonstration, we have "raw, unmediated perception"? Something we can point to and say, "This," sans interpretation? — J

This isssue is formulated in an interesting way by Joseph Rouse, who contrasts his view of discursive normativity as functioning ‘all the way down’ from the accounts of figures such as Quine, Sellars, McDowell, Davidson and Brandon, who each in different ways relies on a sovereign. account of nature to ground statements of fact.

Accounts of the intralinguistic, ra­tional determination of semantic contents and their inferential or other ho­listic interrelations ultimately depend on showing how those relationships are accountable to causal capacities and interactions of worldly objects. Each account—Quine on holistic adjustments of theories at the “tribunal” of sen­sory surface irritations, Sellars on integrating the manifest and scientific im­ages of humanity-in-the-world, Davidson on token identity of mental and physical events, John McDowell on relations between law-governed first na­ture and conceptual capacities inculcated by second nature, or Brandom on judgments of practical and perceptual reliability—fails. The underlying dif­ficulty is their effort to separate rational, normative relations among semantic
contents from their realization by humans as living organisms who evolved and developed in discursively articulated environments.

Yep, as you said there, it's not that conceptual schemes can';t be relative, but that there can't be conceptual schemes. — Banno

And , if I understand Davidson correctly, there cannot be conceptual schemes thanks to what Rouse calls Davidson’s assumption that semantic meaning is grounded in the ‘token identity of mental and physical events,’

One can delimit a measure arbitrarily. This doesn't mean all measures are arbitrary. To count "brown animals" requires knowing an animal, an organic whole, as a unit. I don't think this is arbitrary. For instance, show me a single culture that sees what we consider to be a cow, bear, etc. to be more or less than one of that sort of animal.

The salient bit is that number is a way of thinking about (talking about, treating, approaching) the animals.

Well, the separation, (or inseparability) of thinking and being is a whole rabbit hole. My point would merely be that, when paleontologists unearth two fossilized birds who fell into a tar pit together when the branch they were sitting on snapped 2 million years ago, they (and we) are justified in thinking that there were indeed two birds that fell into the tar pit. This, despite this event being prior to man or any human languages.

Not all wholes are so obviously wholes. There is no doubt a gradient here. But individual animals, planets, etc. make good examples of multitudes.

On a side note, it is strange to me that claims running counter to my own (e.g. "Mars and Saturn being two distinct planets is a fact that is, in an important sense, not dependent on we speak of them.") sometimes feature charges of "anthropocentrism." It seems to me that claims like "there are never two tigers in a clearing, two stars in a binary star system, etc. but that man speak or think of them so," are themselves the height of anthropocentrism, a sort of desiccation of being outside the gaze of man.

My point would merely be that, when paleontologists unearth two fossilized birds who fell into a tar pit together when the branch they were sitting on snapped 2 million years ago, they (and we) are justified in thinking that there were indeed two birds that fell into the tar pit. This, despite this event being prior to man or any human languages. — Count Timothy von Icarus

Did you think that somehow this is incompatible with the account I gave? How?

It seems to me that claims like "there are never two tigers in a clearing, two stars in a binary star system, etc. but that man speak or think of them so," — Count Timothy von Icarus

Again, how would such an oddity follow from the account given? If someone counts no tigers when there are two, they are in error.

I probably misunderstood you then. I took: "number is a way of thinking about (talking about, treating, approaching) the animals" to mean that number only/primarily shows up in our (i.e. human) speech and behavior.

Ok.

being a bishop is a way of treating that piece of wood, being a dollar coin is a way of treating that piece of metal and being two animals is a way of treating that cat and dog. — Banno

My point would merely be that, when paleontologists unearth two fossilized birds who fell into a tar pit together when the branch they were sitting on snapped 2 million years ago, they (and we) are justified in thinking that there were indeed two birds that fell into the tar pit. This, despite this event being prior to man or any human languages.
— Count Timothy von Icarus
Did you think that somehow this is incompatible with the account I gave? How? — Banno

A way of treating something as something is a convention. How can a convention pre-exist the existence of human beings on the planet? It’s one thing to say that there was a world prior to the arrival of humans and our conventions of language, but it’s another to specify the nature of that world (two birds, or a cat and a dog) on the basis of our contingent discursive accounts of it. It is neither true nor false to say that there were a countable number of animals prior to the arrival of humans.

. . . to specify the nature of that world (two birds, or a cat and a dog) on the basis of our contingent discursive accounts of it. — Joshs

Just to keep the argument clear here, what should we say the description "a cat" is contingent upon? Obviously I'm not looking for a reply along the lines of "It's contingent upon language" -- that goes without saying. But what else? What are the factors that suggest that particular bit of language?

Rouse is helpful here in showing this connection among the five philosophers. Whether all their accounts fail, I couldn't say. When he writes, "The underlying dif­ficulty is their effort to separate rational, normative relations among semantic contents from their realization by humans as living organisms who evolved and developed in discursively articulated environments," is this a somewhat awkward equivalent to "There are no propositions that aren't 1st person singular or plural"?

When he writes, "The underlying dif­ficulty is their effort to separate rational, normative relations among semantic contents from their realization by humans as living organisms who evolved and developed in discursively articulated environments," is this a somewhat awkward equivalent to "There are no propositions that aren't 1st person singular or plural"? — J

The way I think he would put it is ‘all propositions, including 1st person plural, are derivative of 1st person singular stances, but the 1st person singular includes within its own autonomy its discursive, partially shared circumstances with others.

Just to keep the argument clear here, what should we say the description "a cat" is contingent upon? Obviously I'm not looking for a reply along the lines of "It's contingent upon language" -- that goes without saying. But what else? What are the factors that suggest that particular bit of language? — J

Language itself is contingent on our material interactions with the world, interactions which constitute a field of possibilities of action out of which objects emerge as what they are. It is our doings that produce such fields. If we are deprived at an early age of the ability to touch, pick up and interact with things, we don’t develop the ability to see them as meaningful objects. We see a visual field, but the meaning of object only makes sense in terms of what we can do with it.

OK. Rödl is dealing with some similar issues in Self-Consciousness and Objectivity. His "absolute idealism" leads him to very different conclusions, of course, but he and Rouse are both trying to supply an account of the given, what is present to consciousness.

Well, this is a pretty general formula. I was hoping you could use "a cat" as an example and describe what the "contingent discursive account" looks like, which allows us to use it to "specify the nature of the world."

Hello again to everyone,

Since it seems that no one other than myself is voting for the answer that @Banno offered as a response to the question of the OP, please allow me to attempt to answer it in my own way. I have already suggested my answer in the preceding pages, but now I will express it in a clearer way.

Do infinitesimals exist (in the platonistic sense)? — Michael

No, they do not. Nothing exists in the platonistic sense, if by "platonistic sense" you mean ideal existence. It can be argued (as Mario Bunge has argued in print) that all numbers, including infinitesimals, are really just brain processes occurring in the brains of living humans. That goes for infinitesimal as well as for the set of the natural numbers. It goes for every mathematical object in general, including the objects of geometry, algebra, arithmetic, number theory, mathematical analysis, logic, and the very foundations of mathematics as such. It's not just a "Do numbers exist?" sort of question.

1. If they don't exist then any number system that includes them is "wrong". — Michael

False. For one can declare that mathematical objects in general, and infinitesimals in particular, have "conceptual existence", as opposed to "real existence", which is precisely what Mario Bunge argues in his book from 1977 called "Ontology I: The Furniture of the World".

2. If they do exist then any number system that excludes them is "incomplete" (not to be confused with incompleteness in the sense of Gödel). — Michael

False. They exist only in a conceptual sense, not in a real sense, as I have just said. They have "conceptual existence", and what that means is that they are just useful fictions in a quasi-Nietzschean sense. This is precisely what Bunge argues. What infinitesimals really are, is a series of processes occurring in the brain of a living human. If you ask Bunge if "there is a number right there" and you point to a visual sign like "3", which you can physically see with your eyes, Bunge would say no, that's not "a number", that's simply a numeral. It's a meaningless visual shape, and we, humans, have agreed to give it a meaning. It means "three". Three what? Three x, whatever x may be. But all of this is conceptual existence. Numbers, understood "like that", as in realistically, are just a series of brain processes, as I've pointed out earlier.

3. Infinitesimals exist according to some number systems but not others. This would be fictionalism, — Michael

False. This is because the entire explanation that I gave before, which is Bunge's explanation, can be accurately characterized as adhering to mathematical fictionalism. Bunge himself sees it that way, and he has manifested that belief in print, in an unequivocal way.

And , if I understand Davidson correctly, there cannot be conceptual schemes thanks to what Rouse calls Davidson’s assumption that semantic meaning is grounded in the ‘token identity of mental and physical events,’ — Joshs

Is that a criticism or an explanation?

It's not a poor description of Davidson's approach. But Davidson's explanation is not an assumption so much as a description of how nouns work. "Joshs" refers to Joshs, but Joshs is the fellow who wrote that post. Asking which of these Joshs is mental and which physical makes as much sense as asking which of them are made of cheese.

But then it is the presumption that things must be either mental, and hence not physical, or physical, and hence not mental, that underpins much of the confusion expressed hereabouts. Descartes legacy.

Added:

A way of treating something as something is a convention. How can a convention pre-exist the existence of human beings on the planet? It’s one thing to say that there was a world prior to the arrival of humans and our conventions of language, but it’s another to specify the nature of that world (two birds, or a cat and a dog) on the basis of our contingent discursive accounts of it. It is neither true nor false to say that there were a countable number of animals prior to the arrival of humans. — Joshs

That last sentence is wrong. There were indeed a countable, but unknown, number of animals in the world at any point in the past. That this is so follows from the number of animals being a natural number that is not zero nor infinite.

So we might well perform a reductio, and see which assumption of the argument is in error. Of course a convention can be used to talk about the past. One Million BC is a year. We can talk about how many animals there were back then. Supposing that doing so makes no sense is a philosopher's conceit.

But individual animals, planets, etc. make good examples of multitudes. — Count Timothy von Icarus

But even these examples are not clean.

When counting cats in the room, are you counting that wildcat? This cat is pregnant, are you counting its unborn fetuses?

When you are counting animals in the room, are you counting the insects? The gut bacteria? The mitochondria? Yourself?

When you are counting planets, are you counting moons? Planetoids? Asteroids?

The reason these are not clean is that categories like "cat", "animal", "planet" are not found in nature. They are invented, not discovered. Which is not to say they are not meaningful. Because categories are subjective creations does not mean they are whimsically chosen. They sometimes have deep ontological bases (i.e. life vs. non-life), and they sometimes do not (i.e. racial categories).

The whole crux of the problem as I see it is that while numbering entails categorization which is a subjective act, numbers nonetheless have very objective properties.

Yep.

To count "brown animals" requires knowing an animal, an organic whole, as a unit. — Count Timothy von Icarus

Rather, to count "brown animals" requires seeing an animal as an organic whole, as a unit. That's a fair bit less than "knowing".

It can be argued (as Mario Bunge has argued in print) that all numbers, including infinitesimals, are really just brain processes occurring in the brains of living humans. — Arcane Sandwich

I'll repeat a simple argument against this.

If π is a brain process in your brain, and also a brain process in my brain, then it is two different things.

But if that were so, when I talk about π I am talking about a quite different thing to you, when you talk about π.

When we each talk about π, we are talking about the same thing.

Therefore π is not a brain process in your brain

I'll repeat a simple argument against this.

If π is a brain process in your brain, and also a brain process in my brain, then it is two different things.

But if that were so, when I talk about π I am talking about a quite different thing to you, when you talk about π.

When we each talk about π, we are talking about the same thing.

Therefore π is not a brain process in your brain — Banno

And here is the Bungean retort to your argument:

Ideas, then, do not exist by themselves any more than pleasures and pains, memories and flashes of insight. All these are brain processes. However, nothing prevents us from feigning that there are ideas, that they are "there" up for grabs - which is what we do when saying that someone "discovered" such and such an idea. We pretend that there are infinitely many integers even though we can think of only finitely many of them - and this because we assign the set of all integers definite properties, such as that of being included in the set of rational numbers. — Bunge, Ontology II: A World Of Systems, page 169)

Seems to me to entirely miss the most important bit; that π is communal.

We don't pretend that there are infinitely many integers, because there are infinitely many integers. That's how integers work. And they work that way not just in this or that mind, but as an activity performed by our community.

We don't pretend that there are infinitely many integers, because there are infinitely many integers. — Banno

Where are they, then? Are they under my table? Maybe some of them are there, I should check. Are they inside a box in my living room? Are they growing in the tree in my back yard, as if they were fruits? You say there are, emphatically. So, I ask you: where are they?

So you are now saying that there are not infinity many integers?

We can quantify over things that are not physical. You appeared to understand this, a few days ago. But it's late in your party of the world.

If π is a brain process in your brain, and also a brain process in my brain, then it is two different things.

But if that were so, when I talk about π I am talking about a quite different thing to you, when you talk about π.

When we each talk about π, we are talking about the same thing.

Therefore π is not a brain process in your brain — Banno

It is largely the very peculiar kind of being that belongs to universals which has led many people to suppose that they are really mental. We can think of a universal, and our thinking then exists in a perfectly ordinary sense, like any other mental act. Suppose, for example, that we are thinking of whiteness. Then in one sense it may be said that whiteness is 'in our mind'. In the strict sense, it is not whiteness that is in our mind, but the act of thinking of whiteness. The connected ambiguity in the word 'idea' also causes confusion here. In one sense of this word, namely the sense in which it denotes the object of an act of thought, whiteness is an 'idea'. Hence, if the ambiguity is not guarded against, we may come to think that whiteness is an 'idea' in the other sense, i.e. an act of thought; and thus we come to think that whiteness is mental. But in so thinking, we rob it of its essential quality of universality. One man's act of thought is necessarily a different thing from another man's; one man's act of thought at one time is necessarily a different thing from the same man's act of thought at another time. Hence, if whiteness were the thought as opposed to its object, no two different men could think of it, and no one man could think of it twice. That which many different thoughts of whiteness have in common is their object, and this object is different from all of them. Thus universals are not thoughts, though when known they are the objects of thoughts. — Bertrand Russell, Problems of Philosophy - The World of Universals

Consider such a proposition as 'Edinburgh is north of London'. Here we have a relation between two places, and it seems plain that the relation subsists independently of our knowledge of it. When we come to know that Edinburgh is north of London, we come to know something which has to do only with Edinburgh and London: we do not cause the truth of the proposition by coming to know it, on the contrary we merely apprehend a fact which was there before we knew it. The part of the earth's surface where Edinburgh stands would be north of the part where London stands, even if there were no human being to know about north and south, and even if there were no minds at all in the universe. We may therefore now assume it to be true that nothing mental is presupposed in the fact that Edinburgh is north of London. But this fact involves the relation 'north of', which is a universal; and it would be impossible for the whole fact to involve nothing mental if the relation 'north of', which is a constituent part of the fact, did involve anything mental. Hence we must admit that the relation, like the terms it relates, is not dependent upon thought, but belongs to the independent world which thought apprehends but does not create.

This conclusion, however, is met by the difficulty that the relation 'north of' does not seem to exist in the same sense in which Edinburgh and London exist. If we ask 'Where and when does this relation exist?' the answer must be 'Nowhere and nowhen'. There is no place or time where we can find the relation 'north of'. It does not exist in Edinburgh any more than in London, for it relates the two and is neutral as between them. Nor can we say that it exists at any particular time. Now everything that can be apprehended by the senses or by introspection exists at some particular time. Hence the relation 'north of' is radically different from such things. It is neither in space nor in time, neither material nor mental; yet it is something.

The above also applies to number.

↪Arcane Sandwich
So you are now saying that there are not infinity many integers?

We can quantify over things that are not physical. You appeared to understand this, a few days ago. But it's late in your party of the world. — Banno

Alright, let me phrase it in communitarian terms, then, to speak your dialect for a moment. The only person (to my eyes, at least) that has attempted, in the last few days, to solve the question presented in the OP, is you. And the only person that you managed to convince, was me. These other fine people here with us in this Thread, are working on their own solution to the problem presented in the OP. As in, you have not convinced them of your solution in that sense, you've only convinced me. So, simply as an act of courtesy towards you, I'm now disagreeing with you. But I do it for two reasons:

1) Firstly, because no one is even challenging your solution to the question of the OP in the manner that I am, and;
2) Secondly, because in that specific sense, my solution is better than yours, because my solution is technically Bunge's solution to the problem. If this is reduced to community terms, I prefer to agree with Bunge than with you on that point. So, you see why there's a problem with the very notion of "community Math" to begin with as a concept. Math has to be absolute, in the formal sense that "it's not up for debate", it's not for the community of mathematicians to decide. Whatever is said in formal languages, such as Logic and Math, has to be said in such a way as to be objective and unambiguous as possible. That cannot be done in ordinary talk, no matter how sophisticated. It must be done formally, in a purely formal language, such as the language of first order predicate logic, or set theory, or some other sort of formal language.

Now, what is the explanation for that? What is the "underpinning" of it, so to speak? It is biology, apparently. As in, it is the biology of the brain of a member of the human species.

So, do numbers exist out there in the world? What exists, at most, is a visual sign, such as this two-stick looking thing that we call "seven": 7

Is that meaningless sign a number? I would say no. That's not a number, that's a numeral. And there are no numbers when you count ordinary objects: there are ordinary objects, and each of them has a "oneness" that makes it an individual object. But that is not Math, and it is not Logic, it is Ontology.

It is biology, apparently. As in, it is the biology of the brain of a member of the human species. — Arcane Sandwich

If a species evolves to the point where it can recognise 'the law of the excluded middle', does that entail that 'the law of the included middle' can be understood as a product of biology?

If a species evolves to the point where it can recognise 'the law of the excluded middle', does that entail that 'the law of the included middle' can be understood as a product of biology? — Wayfarer

Hmmm... what a clever question. Are you sure that this isn't your first rodeo, partner? Let's see.

(spit to the side) The question that you're asking, Sir, is the question that John Dewey asked of the philosophies of both Charles Sanders Peirce and William James. The three of them were Pragmatists, you see, and today they are something like "The Holy Trinity of Classical Pragmatism". Make of that what you will, I just made it up because it sounded pleasing to my ear before I even said such a thing. Not that I take it back, though. Because I do not.

Right, so what did Dewey himself answer? Well, Dewey was of the opinion that, yes, effectively so. If a species evolves to the point where it can "recognize" what we, humans, call "the law of excluded middle", that does indeed effectively entail that "the law of the included middle", as you so cleverly call it in opposition to "excluded", can be understood as a product of biology. And if you simply made a mistake there, intending to say "excluded" instead of "included", that too, dear Sir, can also be understood as the product of biology.

And Dewey held that opinion. What was Bunge's opinion on that matter? I would not know. I'm afraid that no one would, apparently. It seems to be an issue that Bunge himself struggled with as a philosopher and as a scientist, so he was somewhat "silent" or "agnostic" about it.

That by way of agreement? Can we cure @Arcane Sandwich of his reductionism? :wink: