Syntactically: P v ~P

Semantically: Every sentence in the language is either true in the model or it is false in the model (where 'or' is the inclusive or; while the 'but not both' clause for exclusive or is demanded by the law of non-contradiction: ~(P & ~P)). — TonesInDeepFreeze

Alright. And how would you write down Wittgenstein's proposal that we should happily welcome contradictions in mathematics, syntactically and semantically? What sort of axiom would that translate into, in your opinion?

I have nothing to say about that.

Though, while probably not specifically apropos of Wittgenstein himself, one can look up the subject of paraconsistent logic.

And how would you write down Wittgenstein's proposal that we should happily welcome contradictions in mathematics, syntactically and semantically? What sort of axiom would that translate into, in your opinion? — Olivier5

I see Heidegger’s approach here as overlapping Wittgenstein’s. Heidegger explains that in taking something to be the case in a propositional judgement (for instance, S is P) , we are taking something as something within a wider context of pragmatic relevance.

“The most immediate state of affairs is, in fact, that we simply see and take things as they are:
board, bench, house, policeman. Yes, of course. However, this taking is always a taking within the
context of dealing-with something, and therefore is always a taking-as, but in such a way that the
as-character does not become explicit in the act.”

“What is to be got at phenomenally with the formal structures of "binding" and "separating," more
precisely, with the unity of the two, is the phenomenon of "something as something." In accordance with this structure, something is understood with regard to something else, it is taken together with it, so that this confrontation that understands, interprets, and articulates, at the same time takes apart what has been put together. If the phenomenon of the "as" is covered over and above all veiled in its existential origin from the hermeneutical "as," Aristotle's phenomenological point of departure disintegrates to the analysis of logos in an external "theory of judgment," according to which judgment is a binding or separating of representations and concepts. Thus binding and separating can be further formalized to mean a "relating." Logistically, the judgment is dissolved into a system of "coordinations," it becomes the object of "calculation," but not a
theme of ontological interpretation."

What Heidegger is saying is that the sense of S is P is always framed and situated within a wider context. Things are the case or not the case within this wider sense-making space, which is context-sensitive. The bottom line is that the meaningful
sense of S is P is a moving target , and what the LEM does is delimit how much the sense of the meaning of the proposition can vary before it becomes incoherent. At that point we blame each other for misunderstanding the definitions.

The question is, what kind of existence conceptual information has. — Wayfarer

What kind of existence does a material object have? A material existence. What kind of existence does conceptual information have? A conceptual existence. This is all just a matter of words as I see it.

The more I think about it, the less sense that kind of question seems to have.

So, if you want to deny that having a conceptual existence is a material function, then what kind of existence do you imagine conceptual information as having?

'Mathematical realism, like realism in general, holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. In this point of view, there is really one sort of mathematics that can be discovered; triangles, for example, are real entities, not the creations of the human mind.

Many working mathematicians have been mathematical realists; they see themselves as discoverers of naturally occurring objects. Examples include Paul Erdős and Kurt Gödel. Gödel believed in an objective mathematical reality that could be perceived in a manner analogous to sense perception. Certain principles (e.g., for any two objects, there is a collection of objects consisting of precisely those two objects) could be directly seen to be true, but the continuum hypothesis conjecture might prove undecidable just on the basis of such principles. Gödel suggested that quasi-empirical methodology could be used to provide sufficient evidence to be able to reasonably assume such a conjecture.

Within realism, there are distinctions depending on what sort of existence one takes mathematical entities to have, and how we know about them. Major forms of mathematical realism include Platonism and Aristotelianism.

Mathematical anti-realism generally holds that mathematical statements have truth-values, but that they do not do so by corresponding to a special realm of immaterial or non-empirical entities. Major forms of mathematical anti-realism include formalism and fictionalism.' ~ Wiki

The question being, if mathematical primitives are real, in what sense are they real? A common expression is 'out there somewhere', but they don't exist in the same sense that material objects do. It's the 'in the same sense' that is problematical'.

But, the rationalist’s claims appear incompatible with an understanding of human beings as physical creatures whose capacities for learning are exhausted by our physical bodies.' — Wayfarer

I would think this part is unwarranted, in the sense that life is already semantic. Our physical bodies are semantic. The really hard problem to me is not consciousness but life itself. Given life, it was only a question of time for some critter to start recording and decoding information in real time so to speak, as information arises around the organism, information acquired through the senses and analysed through some (originally small) brain. These abilities (primitive eyes etc.) appear very very early in evolution.

From a critter that can look at something, to another critter that can reflexively look at itself looking at something, all you really need is some redundancy in brain power.

At some point in our evolution, there were about a dozen (known) species of Australopithecus, Paranthropus and Kenyanthropus and the likes. One of these most probably gave rise to the fist Homo species (habilis). All these species coexisted in Africa, sometimes in the same place but eating different things. The striking fact is that all of them shared, as a sort "special weapon", an inordinately large brain as compared to their body size.

If some platonic truth is out there, evolution can be seen as a very slow process of coming out of the cave.

Well then why did you say that it had nothing or little to see with the LEM, pray tell?

What kind of existence does a material object have? A material existence. What kind of existence does conceptual information have? A conceptual existence. This is all just a matter of words as I see it. — Janus

The interesting thing is that materiality is already ‘conceptual’ through and through in that the very notion of an empirical object is a complex perceptual construction , an idealization. Furthermore , it is this idealizing abstraction at the heart of our ideas of the spatial object that makes the mathematical
possible. They are parasitic on and presuppose each other.

Yes, but none of that actually says anything about what it could mean for something conceptual to exist independently of the human mind. The odd thing is that it seems intuitively easy to imagine physical stuff have an existence independently of the human mind, but anti-realists and idealists deny that and yet want to grant independent existence to the very things we cannot intuitively imagine to have such an existence.

I have explained more than once already why it is not the case that an otherwise consistent system can be made inconsistent by retracting the LEM. It is not required that I propose couching Wittgenstein into formal syntax and semantics to make that point.

The interesting thing is that materiality is already ‘conceptual’ through and through in that the very notion of an empirical object is a complex perceptual construction , an idealization. Furthermore , it is this idealizing abstraction at the heart of our ideas of the spatial object that makes the mathematical
possible. They are parasitic on and presuppose each other. — Joshs

I don't agree with this; I think 'materiality' as a concept is obviously (by definition) conceptual. But material things are sensed, even animals find themselves in an environment comprised of material entities which, judging from their behavior, they must see much as we do (although obviously they don't conceive of them as material entities).

The math involved in structural engineering have changed overtime. If in one of these changes, them engineers postulated that anything mathematical is both true and false at the same time, as Wittgenstein was effectively (though unwittingly) suggesting, they might have ended building quite a few failed bridges. — Olivier5

There is none of that kind of postulation in calculus, even though from the start the modeling of movement in calculus is fictive in the sense that it is not really movement, but a static equation. All the math involved in structural engineering requires is that it works; that it can effectively model things like tensive and compressive forces and the hardness, strength and flexibility of construction materials.

Things are the case or not the case within this wider sense-making space, which is context-sensitive. — Joshs

Yes to this, a universal rule. Statements and propositions are always made as part of a context, in a very real i.e. local, human sense of who says it, but also theoretically speaking as any statement is to be understood as part of a broader conceptual framework. There's no text in a vacuum. And yes, the truth value of a proposition (often even its meaning) always depends on this context.

The bottom line is that the meaningful
sense of S is P is a moving target , and what the LEM does is delimit how much the sense of the meaning of the proposition can vary before it becomes incoherent. At that point we blame each other for misunderstanding the definitions. — Joshs

This is where you lose me. I do not understand what you mean. That the LEM is not universally applicable, that it has a clear domain of applicability here but not there? If yes then ok, it's consistent with the above about propositions being true in certain contexts and not others.

material things are sensed, even animals find themselves in an environment comprised of material entities which, judging from their behavior, they must see much as we do (although obviously they don't conceive of them as material entities). — Janus

What animals ( and humans) ‘sense’ , once we have removed all the higher level constructions that make phenomena appear for us as self-persisting things in a geometric space-time, is a constantly changing, chaotic flux of impressions. Out of this steaming flux we discern regularities and correlations, not just in the changes happening in our environment, but in the relation between these changes and the movements of our body. An ‘object’ is the product of all these correlations and regularities. Most of what we see at any moment ina spatial object is provided by our own expectations based on previous experience with something similar. We mostly construct the object from memory and anticipation. So the idea of spatial objects is an idealization based on actual experience which is contingent and relative.
It is not a fact that objects persist in time , it is a presupposition, and one which is necessary in order for there to be naturalistic empirical science and mathematical calculation.

But if this point, whatever its merits, has nothing to see with what Wittgenstein was saying, why do you bring it up in this thread?

What animals ( and humans) ‘sense’ , once we have removed all the higher level constructions that make phenomena appear for us as self-persisting things in a geometric space-time, is a constantly changing, chaotic flux of impressions. Out of this steaming flux we discern regularities and correlations, not just in the changes happening in our environment, but in the relation between these changes and the movements of our body. An ‘object’ is the product of all these correlations and regularities. Most of what we see at any moment ina spatial object is provided by our own expectations based on previous experience with something similar. We mostly construct the object from memory and anticipation. So the idea of spatial objects is an idealization based on actual experience which is contingent and relative.
It is not a fact that objects persist in time , it is a presupposition, and one which is necessary in order for there to be naturalistic empirical science and mathematical calculation. — Joshs

It seems obvious there are temporally persistent stable objects both for us and animals. My dog sees his food bowl where I see it. I see him going to it. I see no reason to doubt that animals see the same things in the same locations as we do. The dogs use the steps just as I do. The cat climbs the tree. The bird sits on the branch. You don't observe animals trying to climb steps or trees or perch on branches that don't appear to be there to us. When I throw the ball for my dog he can obviously see the ball, he tracks it as it flies through the air and usually manages to be within a meter or two of it when it hits the ground.

Which empiricism generally resists, on the grounds that humans are born 'tabula rasa', a blank slate, on which ideas are inscribed by experience. — Wayfarer

Those of us who have been around young children know that they come out of their mother's anything but blank. They are who they are and always will be the minute they are born. Probably before.

You made a general statement about it. You made your own claim about mathematics and mathematical logic. And your claim is incorrect. My point about that pertains irrespective of Wittgenstein. Whether people are talking about Wittgenstein in particular, or inclusive of other tangents, it is worthwhile to point out that certain specifics mentioned in context or standalone are in error.

All the math involved in structural engineering requires is that it works; that it can effectively model things like tensive and compressive forces and the hardness, strength and flexibility of construction materials. — Janus

Yes yes yes but all this assumes that if the tensive strength of this material is X, it is X. It is not something else than X. There's only one correct value or range (+ or - whatever residual error).

This is where you lose me. I do not understand what you mean. That the LEM is not universally applicable, that it has a clear domain of applicability here but not there? If yes then ok, it's consistent with the above about propositions being true in certain contexts and not others.
3h — Olivier5

Yes, that’s pretty much what I mean. I think the issue here is that the later Wittgenstein seems to be wanting to radicalize the notion of context such that it no longer allows for categories of use. That is to say , he seems to want to make every situation , for every individual, it’s own context. In doing so , he rejects the coherence of rules, grammars, criteria as categories with any existence outside of particular situations and seen from
particular individual perspectives. So , in sum , if one understands context in a formal categorical sense, then the LEM is applicable in some contexts
and not in others. But if one equates context with absolute situational and perspectival contingency , then the LEM can no longer find the minimal categorical identity over time in the idea of context necessary for it to contribute anything useful.

You made a general statement about it. You made your own claim about mathematics and mathematical logic. And your claim is incorrect. — TonesInDeepFreeze

What claim are you talking about?

It seems obvious there are temporally persistent stable objects both for us and animals — Janus

Most of what you say is based on what seems obvious to you.

Well, sure if structural engineers started entering random numbers into their equations there those equations would not yield workable results.

The one we talked about:

[...] allowing contradictions in math is equivalent to dropping the law of the excluded middle from mathematical logic [...] — Olivier5

And I mentioned it most presently only about an hour ago and as you responded to me right after.

You say some ridiculous things sometimes! Do you deny that it seems obvious that there are temporally persistent objects? Is the door always where you expect it to be or somewhere else? Your front steps? Your driveway? Your car? Do you sometimes argue with your wife because you see the car in the driveway and she sees it in the street? When you open the door for the dog and say "In" does he sometimes try to jump into a car that you can't see?

How the fuck would we all function in the world if there were no persistent objects?

When I throw the ball for my dog he can obviously see the ball, he tracks it as it flies through the air and usually manages to be within a meter or two of it when it hits the ground. — Janus

I haven’t read up on this , but if your dog watches you attach something to the side of an object he can’t directly see, will he know that there is another side of the object and look for it there? Will he discern the general
direction and distance of the ball from watching the arc and force of your throw? Research says yes, dogs are capable of object permanence, and cats may also have this ability, but not generally lower animals. Whether he has these skills or not, my point is that it is these and many many other kinds of correlations that determine the very meaning of ‘object’ for an animal. Of course, in many ways a dog is quite close to a human in intelligence , but some differences are obvious.Most Human facial expressions are meaningless to a dog , as is a pointing feature. Now, these are admittedly meanings are are much more sophisticated than those involved in recognizing and tracking objects. But I think you will find all kinds of subtle and not so subtle differences between rabbits and birds and snakes and fish in terms of how ‘objectness’ functions for them.

I have Carpet Pythons on my property. They get into my chook tractor and wait in the nesting compartment for rats. They can only get in because spaces in the steel mesh are large enough. The snakes obviously see the chook tractor where I see it, and are obviously constrained by its temporally persistent physical characteristics. I've never seen a python in there that would obviously be too big to squeeze through the gaps in the mesh.

Also it's not true that dogs can't recognize the act of pointing; in fact they are, if I remember correctly, the only animals that can. They may not read facial expressions, but they certainly respond to differently to different vocal tones.

It beats me why people want to deny that there is a physical environment that we share with others, as well as with other kinds of physical entities, who perceive pretty much the same persistent features of the environment as we do.

Yes, counting is something we do, not something we discover.

Its a way of talking about stuff. The way of talking is made up. The stuff isn't.

There's no need for ghosts. If mathematicians are discovering anything, it's the consequences of a game that started with triangles and numbers.

My suspicion is that the subjective/objective bugaboo is lurking here. That mathematics is made up does not mean it is subjective, private, only in one's head. It's a shared game.

As far as I know, Euclid made mathematics a shared game with his Elements. Much of which was expressed as propositions.

Is there such a thing as ‘pidgin Wittgenstein?’

The point I’m labouring is that there are real intelligible objects. They comprise an enormous range - not of phenomena, because phenomena are ‘what appears’. They inhere in the domain of logic, geometry, laws and conventions - and so on. They’re real, in that they’re the same for all who think, but they’re only graspable by a rational intelligence. Hence they’re ‘intelligible objects’, real to all who are capable of grasping them, but not the product of your or my mind so, not ‘made up’.