Does logical structure entail ontological commitments about things like grounding, “simples,” existence, Ǝx, and other tools of the trade? — J
About the only discussion I'm aware of that elucidates this distinction (albeit in relation to universals rather than number per se) is in Russell's Problems of Philosophy — Wayfarer
What's more, there are, or have been, human languages -- and thus functioning human communities to speak them -- that only have "1, 2, many". So language doesn't directly lead to mathematics more advanced than crows and infants possess, even if it enables it (as it does, you know, everything). — Srap Tasmaner
I don't see how an account that is social practice or activity "all the way down," is going to work. — Count Timothy von Icarus
In Plato these levels or kinds of knowledge were distinguished per the Analogy of the Divided Line . Those distinctions are what have been forgotten, abandoned or lost in the intervening millenia due to the dominance of nominalism and empiricism. — Wayfarer
This is pretty clearly a case in which one language has in its domain a thing which is a compound of this pencil and your left ear, and the other does not. — Banno
I don't want to get embroiled in this thread — Leontiskos
The problem here is that quantification derives from the meaning of 'being' or 'exists', — Leontiskos
There is a longstanding dispute over the univocity of being (and predication) between the Thomists and the Scotists beginning in the Medieval period. The Scotists held to univocity (and Heidegger's first dissertation was on this topic, on a text then believed to be Duns Scotus'). — Leontiskos
Like Macbeth, Western man made an evil decision, which has become the efficient and final cause of other evil decisions. Have we forgotten our encounter with the witches on the heath? It occurred in the late fourteenth century, and what the witches said to the protagonist of this drama was that man could realize himself more fully if he would only abandon his belief in the reality of transcendentals. The powers of darkness were working subtly, as always, and they couched this proposition in the seemingly innocent form of an attack upon universals. The defeat of logical realism in the great medieval debate was the crucial event in the history of Western culture; from this flowed those acts which issue now in modern decadence. — Richard Weaver, Ideas have Consequences
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
Can you state just why you think that incompatibility obtains? — Janus
As for the unfathomable subtlety of living organisms, I'm all for it. I think many things we describe as 'instinct' are impossible to fathom, but that's a completely separate issue. — Wayfarer
They should begin with Plato and only descend to Russell if they feel the need. — Leontiskos
Can you state just why you think that incompatibility obtains? — Janus
So instead of questioning why it is we can understand numbers, how about interrogating the claim that we are, in fact, 'physical creatures whose capacities for learning are exhausted by our physical bodies?' Or is that such an important principle in our 'best epistemic theories' that it has to be saved at all costs? That seems the point of the sophistry of the 'indispensability argument'. — Wayfarer
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 (i.e. number) 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?
I have asked him to explain what could be meant by saying that numbers are real — Janus
the fact that number is real — Janus
by saying that numbers are real beyond our recognition of number in the world — Janus
The dualist will say that they are abstract objects (not spatial, not temporal, causally inefficacious). — Lionino
I don't think quantifiers have much of anything to do with existence or being or any of that. They're entirely about predication -- classification, categories, concepts. Quantifiers are about what things are, not that they are. — Srap Tasmaner
that you are saying there are two realities—the physical ("sensable") and the mental (abstract) which is basically dualism. — Janus
I actually find it odd to hear you say that quantifiers do not implicate existence (real or imagined). — Leontiskos
You'll never teach the concept of prime to a Caledonian crow — Wayfarer
There are simple algorithms for determining whether a number is prime; it's a mechanical process that doesn't require what you call "rational insight — Srap Tasmaner
Frege accepted the traditional rationalist account of knowledge of the relevant primitive truths, truths of logic. This account, which he associated with the Euclidean tradition, maintained that basic truths of geometry and logic are self-evident. Frege says on several occasions that such primitive truths - as well as basic rules of inference and certain relevant definitions- are self-evident. He did not develop these remarks because he thought they admitted little development. The interesting problems for him were finding and understanding the primitive truths, and showing how they, together with infer- ence rules and definitions, could be used to derive the truths of arithmetic.
There are simple algorithms for determining whether a number is prime; it's a mechanical process that doesn't require what you call "rational insight — Srap Tasmaner
Machines are artefacts, are they not? — Wayfarer
I would like to believe that this position is nearer to Kant’s transcendental idealism. There’s no way I posit anything like Descartes ‘res cogitans’ or the seperatness of mind and body. — Wayfarer
↪Janus You always argue from an unquestioned empiricism and can’t see how anything that challenges that can ‘make sense’ in your terms. — Wayfarer
But leaving that to one side, isn't it enough that we want to share the six fruit equally amongst the three of us, to explain the need for counting?
Many people will ignore that too because they will say that numbers aren't real (Field, Azzouni).
I, personally, think mathematics is an empirical endeavor. — Lionino
The dualist will say that they are abstract objects (not spatial, not temporal, causally inefficacious).
— Lionino
Yes, and unfortunately, we have no idea what it could mean to be such an object, apart from, as I said above, it being thought by some mind. — Janus
number would be a real attribute of objects — Janus
If diversity, sameness and difference are acknowledged as being real — Janus
number would be a real attribute of objects
but the numbers themselves would only be real as ways of thinking and dealing with objects, and also as elements in formalized systems of rules elaborated upon that basis.
I mean, of course they implicate it, in the exact sense that they presuppose it -- but they don't have anything to say about it. Rather like the status that "truth" has in logic ... (Existence being not a real predicate, and in any given language neither is "... is true" -- need the metalanguage for that.)
What's asserted in an existentially quantified formula is not really, say, "Rabbits exist," but the more mundane "Some of the things (at least one) that exist are rabbits." Or "Not all of the things that exist aren't rabbits," etc. — Srap Tasmaner
Also I always think it's worth rememembering that Frege's quantifiers, and the rest of classical logic so many of us know and love, was not designed as an all-purpose logic at all, but was what was needed to formalize mathematics. It's got some very rough edges when applied more broadly, about which there's endless debate, but it runs like a champ on its home turf. — Srap Tasmaner
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