I don't believe that intelligibility can extend to fundamentality. So, whatever names we use to denote it: substance, God, the Real, Firstness, the noumenal, the Will, the Apeiron, Buddha Nature and so on, will, with all their associations and connotations, be tools to relate them to our various systematic understandings in the intelligible world, the 'World as Idea' as Schopenhauer calls it. — Janus

My argument is that intelligibility can approach it in the limit - as its own "other". So intelligibility can define the unintelligible as that which it is ultimately not.

And because we know intelligibility to exist, then we know that - whatever else - its unintelligible ground had to contain intelligibility as its potential. So we can actually know something usefully definite about fundamental unintelligibility.

This is apophatic reasoning. But hey, in metaphysics that is unreasonably effective. ;)

So, the idea of tychism is really just a dialectical negation of the idea of regularity, stability, concreteness; in short of 'being something'. — Janus

Exactly. We recover the pre-dialectical through dialectics itself.

Your talk about the fundamental is just dialectics. If we and the Cosmos are an effect, therefore there was a cause. If we and the Cosmos are emergent, then something was the more fundamental.

So the question of creation and being is always going to be dialectical and apophatic. You then need to scout around the history of metaphysics and see who does the job the most rigorously in this regard,

Spinoza's substance was not thought by him to be "anything", but more like being everything and nothing, inasmuch as to be anything is to be a mode of substance. Hegel similarly said that pure being is close to being pure nothingness.We find apophatic notions of God or Buddha Nature that can be traced back thousands of years. So we can say of Tychism, as Hegel says in another context, that it is the "same old stew reheated". — Janus

I've always said this same old stew has been on the back-burner since the dawn of metaphysical thought. I give full credit to Anaximander with his system of apeiron and apokrisis.

And having checked out many thinkers, Peirce just keeps surprising me with the completeness of his approach. He sorted it out at a fundamental logical level with his triadic model of development. He put the intelligible into the intelligibility.

If you can point out a defect in his analysis, have at it. But telling me others said similar things is not a criticism, is it? My claim is he said it best.

or, on the other hand, that it is the emanation of an unfathomable, infinite intelligence — Janus

So the great unintelligible intelligibility that blindly chose? Does posing an actual contradiction as the origin of being help your case?

Is talk of "emanation" not just hand-waving dressed up in a fancy word?

he point of this is that the emergence of concrete somethingness as a process cannot be intelligibly traced back into firstness, because that is where intelligibility ends. We cannot say what is the symmetry of firstness that is broken to produce secondness, unless we impute an intelligence (albeit of an unfathomable order) to firstness, an intelligence of which our intelligence is a temporal reflection. — Janus

That just isn't a logical argument.

If firstness is where intelligibility ends, then intelligibility is (apophatically) defining it. And for me - given that my worldview is based on the emergence of constraints - apophatic is good. It is fundamental itself.

But you are having to resort to paradox and self-contradiction. You have to talk about intelligences that are unfathomable. You are having to talk about complexity - rational structure - being actually present when there is meant to be only a state of fundamental simplicity.

I just don't get how you can prefer blatantly self-contradicting positions when the alternative is so logical and elegant.

That is the limitation of Schopenhauer's system; it is inexplicable that an ordered Cosmos can be the expression of a blind will. The same goes for any system that thinks firstness as a blind chaos. — Janus

Well yes. A blind intelligence is a nonsense. But a blind chaos isn't. It would seem definitional of chaos that it lacks rational structure. So again, I'm just not understanding how you can really believe your own line of argument here.

I get that you are psychologically committed to some notion of a creating God. But so far you are not revealing any hole in a Peircean process philosophy perspective.

Creating gods seem necessary to a certain brand of logic - the one that believes in mechanical or concrete chains of cause and effect. But that is the very logic that leaves out formal and final cause so as to describe the world solely in terms of material and efficient cause. The logic leaves the blank - the absence of formal and final cause - that a creating intelligence then "naturally" has to fill.

So really, in my view, you are just responding to an obvious hole in reductionist cause and effect thinking. It leaves out formal and final cause right from the beginning. So formal and final cause is what you know must be jammed right back in that blank slot.

But Peirce - and all the other systems thinkers and natural philosophers since Anaximander - have a larger dialectic understanding of logic. Formal and final cause have their proper place in the metaphysical system. The blank space left by reductionism is filled by the logic of holism.

And now you don't need some purposeful and transcendent creator. The Comos can spontaneously self-organise out of pure possibility. With Firstness or Apeiron, there is just nothing to prevent that happening, and so it does.

And retrospectively, the outcome will be judged optimal. The Cosmos might start out trying to express every concrete option, but then with all the options in self-competition, the variety of ways of being will be reduced to the outcome that proves the most effective (at being enduring and continuous - or synechic as opposed to tychic in Peirce's jargon).

I don't see how you can deny the simple logic of this application of natural selection to cosmic evolution. Scientific cosmology is now based on this very metaphysics - the Big Bang as a collapse of the universal wavefunction. So it is not as if we lack physical evidence for it. Quantum mechanics tells us classical reality is emergent from a "sum over histories" or path integral. Even a particle gets from A to B by "taking every possible route", and then the actual route is whatever turns out to be the "least action" or energy-optimising path.

So logic tells us the right kind of metaphysical answer. Philosophy of science tells us why reductionist science left the feeling of there being a blank so far as formal and final cause are concerned. And now modern science itself has filled in that blank (or is trying to) by a holistic model in which classical reality emerges from naked potentiality coupled to natural selection.

In the face of all this, you still prefer paradoxical stories about unfathomable intelligences, blind choosing, and "emanations"? Does that really sound like strong metaphysics?

And remember where this started - your claim that abstract thoughts are biochemical processes. You followed that howler by jumping the other way - saying the mind was in no way the product of informational processes. — apokrisis

The way this started IIRC is that you accused me of being a dualist and have then proceeded to make a dualist argument for the past several posts.

I actually haven't got much to say at this point and would like to wind this down. I do appreciate that this convo has taken a turn for the civil and interesting. But I'm really argued out on this topic at the moment. I will take your advice and go read up on semiotics. It wouldn't be the first time I've taken a run at that subject but I do admit it eludes me.

But if thought isn't biochemical then what is it? Oh, "informational." But you are using that word in a specific way that's different than what I mean by it.

This second misstep was based on your very narrow conception of information processing - one rooted in TMs. — apokrisis

Ah yes. Right, we agree on our point of divergence. Well, information processing is a TM. That's the technical definition. You have a more philosophical orientation. You said a post or two back that "information is meaning." I could not disagree more. So we have identified another point of disagreement.

The reason for the unreasonable effectiveness of TMs is that they are the theoretical limit on semiotic encoding. Semiosis depends on symbols. A TM is the conceptually simplest — apokrisis

I must say I actually find that comment interesting. I promise to do my homework on this. The theoretical limit on semiotic encoding. I don't think I understand entirely what it means, but I do find the idea thought provoking. I'll do some thinking and reading.

This ignores the fact that the flying machine designers quickly gave up trying to copy the flapping wings of birds and instead focused on a non-bird model of flying machines. The flapping did not prove "unreasonably effective".

Whereas the opposite is the case with NNs. Having got programmable computers, it was the case that even just emulating biologically-inspired information processing architectures was "unreasonably effective" for certain tasks, like pattern matching. — apokrisis

But that doesn't prove that minds work that way. Only that NNs have been doing some amazing things. Ever since Deep Blue beat Kasparov I've had to pay attention to weak AI. It's impressive. But I'm sure you know that chess playing algorithms operate very differently than chess playing humans. So my point about birds and airplanes stands IMO.

So that is a particularly inapt comparison with which to make your case. — apokrisis

No, it's perfectly on target. In the beginning, we tried to program chess algorithms with expert knowledge. (You remember the expert systems movement I'm sure). That got the algorithms to a certain level. But to achieve mastery of the game, the designers gave up trying to teach the machine strategy. They just turned the NN loose and let it train itself. [That's the Alpha Zero approach]. So algorithms play chess very differently than humans do. The bird/airplane analogy is directly on point.

So an organism is a machine? You seem out of touch with biology. — apokrisis

Well if it's not, then you are making a dualist argument. Because if something is not physical, then what is it? You'll say "informational" and then I'll point out that the mind isn't a TM and you'll say that information processing is not a TM and I'll say you're wrong about that.

Are you making a dualistic argument or not?

Artificial Life Needs a Real Epistemology - H. H. Pattee
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.18.1316&rep=rep1&type=pdf — apokrisis

Thank you, I'll read that.

So on the one hand you can't even define what you might mean by mind. On the other, you can make confident claims about neuroscience having a quite limited understanding. — apokrisis

I said no such thing about neuroscience. I objected to your claiming that neuroscientists think the mind is a computer program. Which is the same exact thing as an "informational process" even though you keep claiming it isn't.

And you keep reverting to talk of "brain biochemistry" when the question is about cognitive functions. — apokrisis

And physicists keep talking about "gravity" when the question is about falling functions.

If cognitive functions aren't a product of our physical bodies, then you are a dualist. Which is fine, except that we got started when you accused me of being a dualist.

Don't you see the inconsistency of one minute admitting to knowing little, the next to be making a sweeping judgement of the whole field? — apokrisis

You're the one who made a sweeping statement about neuroscientists. I pointed out that at best, SOME neuroscientists agree with you and others don't.

That's my understanding of the Church-Turing thesis. If you have a different idea I'd be interested to hear it.
— fishfry

That defines computation in the general limit ... if you are computing number theoretic functions. — apokrisis

Number-theoretic functions can implement everything else as far as we know. After all a NN is just a set of nodes with numeric weights. An NN is just a set of number-theoretic functions. Turing understood that number-theoretic functions aren't a limitation; but that rather they can be used to compute anything that we think of as computable.

So perhaps brains might not be that kind of "computer". — apokrisis

I'm perfectly willing to grant you that. I even agree with you. But then what kind of computer are they? What kind of computer ISN'T "that kind of computer?" The answer requires a breakthrough in physics and computer science.

Maybe there is not a single arithmetic operation involved in their neural processes. Maybe even "summing weights" is just an analogy for the integrative processes of brain cells. — apokrisis

Well summing weights is not an analogy for how NNs work. It's the literal truth of how they work. And by admitting that when it comes to brains, NN's are at best an analogy, you are conceding my point. Brains aren't NNs. You just agreed that they're only analogies to NNs.

Church-Turing may have zilch to do with neurology. — apokrisis

Right. May have. Some future genius is going to have to sort this out because as of today, if something is a computational or informational process, it's essentially a TM. If minds are informational but not TMs, then science has no current understanding of what that would be.

And yet it is still wrong to then attribute neural information processes to "biochemistry". — apokrisis

You are quite the dualist. If something's not physical, what is it?

And how could you have a view either way without a little more neuroscience to inform your opinion? — apokrisis

How could you have a view either way without a little more computer science to inform your opinion?

Given that TMs require no more physics than a gate that can read, write and erase a symbol on an infinite tape, why the heck would we expect new physics to make a difference to Turing universal computation? — apokrisis

Because we have to go past the Church-Turing thesis to find a mode of computation that's not a TM, and that has the ability to implement mind. And I believe that we'll need new physics to make that breakthrough. I have some ideas along this line but I don't want to toss them out at this moment.

The power of Turing machines is that they need the least physics we can imagine. What more do you want - time travel, Hilbert space, quantum teleportation? — apokrisis

Mind. Isn't that the "hard problem" as they say? We're trying to figure out how a body can have a mind. It's real puzzler.

That's back to front. It is the virtual elimination of any complicated physics which is the guarantee of the computational universality. — apokrisis

If brains implement minds and brains aren't TMs then what exactly are minds? Isn't that the question? I don't have the answer but nobody else does either. Not even the neuroscientists, who you seem to think have already settled the matter.

When you say "information is meaning," that's something I absolutely deny by my definition of information.
— fishfry

Who could win an argument against your private definitions? — apokrisis

Private between me and every computer scientist and information theorist in the world. Information is a bitstring that can be cranked out by a TM. Randomness is a bitstring that can't. Kolmogorov. He and I inhabit the same private world, although he's a hell of a lot smarter. And deader.

So let's stick to the real world of science, maths and philosophy. — apokrisis

I hardly regard math as part of the real world. it's literally the opposite of the real world. And philosophy has even less of a claim on being real. Surely you know this.

If you want to talk about Shannon entropy, fine. But then we all know that is based on counting meaningless bits. If we understood the pattern to mean something, then each successive bit would fail to be such a surprise. — apokrisis

If the next bit's not a surprise, it's because there's a TM behind the curtain. Which reminds me a little of the intuitionists, who think there's some kind of active agent creating "free choice sequences." I tried to understand that idea once and gave up. Is that what you are referring to here?

If I know you are transmitting the digits of pi, I could stop you right after you said "3". — apokrisis

Of course. Pi is a computable number. We only need to write a program to implement any of the many known closed-form expressions to calculate its digits. You know you're drifting into agreeing with me again.

You don't get it. — apokrisis

LOL. By which you mean, "You have an opinion that differs from mine, and I haven't got much of an argument."

Information theory defines a baseline where the meaning of a bit string is maximally uncertain. — apokrisis

Um ... that's a little murky.

Each bit says nothing about the following bit. — apokrisis

Yes ok.

Then from that baseline, you can start to quantify the semantics. You can derive measures such as mutual information that speak to the information content. — apokrisis

Murky. Bitstrings generated by TMs we call information; and those not generated by TMs, we call random. Quantifying semantics is a mystifying phrase to me. I'm not saying it's meaningless or wrong, just that I don't know how to quantify semantics or what that would mean.

That's one of the advantages of a semiotic approach to the whole issue. It recognises that there is a modelling relation involved. A symbol has meaning due to a habit of interpretation. — apokrisis

The habits of the human minds that assign meaning to the symbols. And even that is a function of language. A cat is a gato is a chat. Meaningless strings of symbols, assigned meanings by humans.

That habit is tied to action in the world. — apokrisis

Actions of humans.

So the informational side of the equation is causally connected to the material side. — apokrisis

Causally? No. The link seems arbitrary. As George Carlin asked, why do we park in the driveway and drive on the parkway?

The link between words and their meaning is anything BUT causal. It's arbitrary. Isn't it?

There is only meaning in relation to the material consequences of any beliefs. — apokrisis

Murky. You were making the claim that there's a causal connection between the arbitrary string of symbols "gato" and a furry four-legged domesticated carnivorous mammal with soft fur, a short snout, and retractile claws. You're wrong about there being a causal relation. But this quoted sentence seems to come out of nowhere and mean nothing.

.
Again, read Pattee - http://www.academia.edu/3144895/The_Necessity_of_Biosemiotics_Matter-Symbol_Complementarity — apokrisis

I promise to do so. I have a lot of research and thinking to do as a result of this thread.

You keep misrepresenting my argument. — apokrisis

I'm shocked. You have an argument? Sorry couldn't resist. You keep misrepresenting my argument too. But I think we're really repeating ourselves at this point. I really have to stop responding.

The significance of an NN would be that it captures something important about brain cognition. That is different from claiming the brain is literally just an NN. — apokrisis

Well then you have backed off your claims and essentially agreed with me. I am in full agreement that NNs have done some amazing things in constrained domains. That does NOT mean the brain is an NN. Please don't make me say birds/airplanes again.

And you seem confused about algorithms. — apokrisis

LOL. That's like me saying you're confused about Peirce. It would push your buttons only for the sake of doing so.

I'm not confused about algorithms. I think very clearly about algorithms. That's why I know that algorithms are dumb as rocks. And that's an insult to rocks.

They are rules for making calculations. So they are something we think it meaningful for a TM to do. They are not the barest syntax of rule following we can imagine. They are semantic actions performed on a machine. — apokrisis

Oh My God. No. TMs are not semantics. I'm sure you didn't mean to write that. I hope you didn't. Algorithms are syntax. Their semantics comes from humans. Please tell me you understand this.

So already we are into the real world where computation carries extra semantic baggage. — apokrisis

Not in my world. Not in anybody's world. Computations are bit flipping. Pure syntax. The meaning is entirely supplied by humans. Remember the 80s movie War Games? The computer doesn't know the difference between playing a game and blowing up the world. To the bit-flipping circuits there is no difference.

The algorithms are intended to represent some actual informational process. — apokrisis

The intention comes from the humans. They are "intended." But the intention appears nowhere in the code.

This could be just handling a company's payroll or driving a video display. Or it could be an attempt to mimic the connective behaviour of neural circuits. — apokrisis

It's only flipping bits. The meaning is in the intention of the human programmers. Play a war game or blow up the world. The computer just flips the bits, it doesn't know the meaning of anything. I really hope you are not confused on this point. I have no idea why you are trying to say something that you must know to be false.

A TM is just a universal algorithm runner. — apokrisis

To be picky, a UTM is a universal algorithm runner. A specific TM only runs one algo. Minor quibble.

How we then exploit that is down to the kind of information processing we think might be meaningful. We have to write an algorithm that seems to perform the task we have in mind. — apokrisis

we have in mind

Yes. You agree with me totally. We supply the meaning. The meaning comes from our minds. There is no meaning and no mind in an algorithm or in a computation.

That could be representing brain functions. — apokrisis

Could be. Yes. Agreed.

It could be representing accounting functions or moving image functions. Universal Turing machines have zilch to say about whether we humans are choosing to run usefully realistic routines or just scrambled garbage randomly concocted. — apokrisis

Right. Why do you so strenuously agree with me?

You are confusing yourself in jumping so interchangeably between talk of TMs, information, computation and algorithms. — apokrisis

No. I'm not confused. An algorithm is a description of a computation. A computation is a physical implementation of an algorithm. Information consists of bistrings cranked out by programs.

Again, we write the algorithms. They have zilch to do with the universality of TMs. So you can't claim them as "mere". They are intended to represent some meaningful relation expressed as some mathematical operation. They have to perform a function we find useful. Thus they could model a company's payroll, or model the cognitive operations of a brain. — apokrisis

Well algos only model a very limited subset of brain activity. And even if you had (in the future) an algo that could simulate the physiological functioning of a brain, it still would not implement a mind. Just as a computer simulation of gravity does not attract nearby bowling balls.

A payroll model is probably pretty ho hum. But a workable brain model? — apokrisis

Just a matter of time an neuroscience. But you are conflating brain with mind. I fully agree that one could in principle someday have a computer program that performs a reasonable simulation of a brain. I deny that such a simulation would have a mind. I agree that one could disagree with what I just said.

Yes, the map is not then the territory. As someone pushing semiosis - a modelling relations view of "information processing" - you don't have to explain that to me. It is what I've been saying. — apokrisis

Well we have been in agreement for quite some time I think.

You are convincing me of your utter unfamiliarity with neural networks in practice. Or even in theory. — apokrisis

In other words you have no argument so you toss out a little gratuitous snark. It's ok, it's the best you can do.

In fact it is completely custom hardware. It is not a simulation of a neural net on conventional technology. It is a direct hardware implementation of a neural network. — apokrisis

It's still conventional computing. Still a physical implementation of a TM. If your friend says he has a computer whose code and/or data can't possibly be backed up, he's lying. What happens when there's a power failure? He types it all in again? I'm aware that there is custom hardware to implement NNs. But it's still based on the conventional model of computation.

Yes, I've spent 40 years being critical of the over-blown claims of computer science. — apokrisis

As have I. I read Dreyfus's What Computers Can't Do many years ago and it's influenced my thinking about all this.

So why are you so hell-bent on convincing me that I'm wrong when I am in full agreement with everything you say except for your tragic misunderstanding of computer science?

So I am basically skeptical of the usual talk of getting close to building "a conscious machine". I know enough about the biology of brains to see how far off any computer system still is. — apokrisis

Which is exactly what I've been saying all along.

Indeed, I would like it if there was an in principle argument for why no mechanical device could ever simulate the necessary biological processes. It would suit my prejudices. So I am just being honest when I confess that there isn't an absolute argument. The effectiveness of NNs suggests that some level of mind-like technology - as good as cockroaches and ants - may be feasible. — apokrisis

I see no disagreement between us here.

And remember where this started - your claim that abstract thoughts are biochemical processes. — apokrisis

It started with you calling me a dualist and then pushing dualistic ideas yourself.

You followed that howler by jumping the other way - saying the mind was in no way the product of informational processes. — apokrisis

It's not, if by informational processes we take the standard contemporary definition in computer science.

This second misstep was based on your very narrow conception of information processing - one rooted in TMs. — apokrisis

If you have a better idea, and not just handwavy misunderstanding, I would like to hear it. If you have figured out how to break the Church-Turing barrier it will make the news because it would refute an 80 year old orthodoxy.

The reason for the unreasonable effectiveness of TMs is that they are the theoretical limit on semiotic encoding. Semiosis depends on symbols. A TM is the conceptually simplest machine for handling symbol strings. — apokrisis

Actually there are much simpler models such as finite automata. But your statement about the limit of semiotic encoding, that's something that I have to go find out more about. It sounds interesting.

A DNA strand can code for a pretty vast array of protein molecules, but that’s it really. Human language can code for a vast array of ideas. That's really powerful as we know. But a TM can implement mathematical algorithms. It can articulate any mathematically-constructable pattern. That is a whole other level of semiosis. — apokrisis

I could write a program that would simulate everything we know about DNA (assuming I took the trouble to learn about DNA). In fact it's the informational basis of life that is the strongest evidence for the point you're trying to make. The next leap is to see how perhaps mind might actually be informational yet not a TM. Or perhaps we're all just programs. It's possible. I don't think so and I don't want to think so, but maybe we're all characters in a Philip K. Dick novel who wake up to discover we're just robots.

So yes. TMs are

really basic. They represent pure syntactic potential, stripped of all physical constraints as well as all semantic. — apokrisis

Yeah yeah. We've both run out of stuff to say.

But then we do have to build b

ack the semantics - add the algorithmic structures - to make TM-based technology do actually useful things. Much like DNA has to code for the kind of neural connectivity that can do actually useful things for organisms. — apokrisis

DNA is the programming of life. But is it the programming of mind? I don't think that discovery's been made yet.

Semiosis recognises the essential continuity here. It sees the ontological difference that codes or syntax makes, the new "unphysical" possibilities they create. — apokrisis

Unphysical. So we're dualists again?

Maybe that's the "physics revolution" you are talking about. I certainly think that it is myself. It explains the information theoretic and thermodynamic turn now happening in fundamental physics I would argue. — apokrisis

The revolution IMO is to break the Church-Turing barrier. We have to find a mode of computation that lets us say that mind is a computation but not a TM. It's my belief we'll need new physics for this.

The way this started IIRC is that you accused me of being a dualist and have then proceeded to make a dualist argument for the past several posts. — fishfry

There's a difference between substance dualism and my dialectical or semiotic approach.

Well, information processing is a TM. That's the technical definition. — fishfry

If that were true, computation becomes a physical impossibility. The technical definition requires the physical manipulation of an infinite tape. Those are quite hard to come by in the real world.

So yes, we can pretend. We can build pseudo-TMs that paper over this embarrassing fact with virtual machine architectures. All you have to worry about now is what you do when you exhaust 64-bit memory addressing, and then 128-bit, etc.

Again, you speak so confidently about neuroscience and computer science. But yet you seem to confuse theoretical constructs with real world practicalities. As was my point, TMs define an ideal limit. And thus a literal TM is also physically unrealisable.

But that doesn't prove that minds work that way. Only that NNs have been doing some amazing things. — fishfry

I can't take you seriously when you make such weak arguments. Of course it is evidence that we are getting at something central to the functional design of the brain. Just as being able to mechanise bird flight would have been evidence we were capturing the essence of the way birds fly.

And just as we instead built fixed-wing planes as the unsubtle brute force alternative, so computers are the familiar clunky von Neumann architectures they have been since computing got properly started. There is nothing biologically-inspired about the design. Yet they do the job - given our limited practical purposes of just getting around or automating various tasks.

In the beginning, we tried to program chess algorithms with expert knowledge. (You remember the expert systems movement I'm sure). That got the algorithms to a certain level. But to achieve mastery of the game, the designers gave up trying to teach the machine strategy. They just turned the NN loose and let it train itself. — fishfry

Yeah. I do remember. And NNs were pushed before that. In the beginning, more naturalistic architectures were being suggested. Check out cybernetics. But then symbolic processing became the 1980s fad. Lisp machines and all that. I happened to edit a computer journal at the time expert systems were getting hot and hyped. The history of all this is familiar.

I objected to your claiming that neuroscientists think the mind is a computer program. Which is the same exact thing as an "informational process" even though you keep claiming it isn't. — fishfry

Stop misrepresenting me. I didn't say neuroscientists think the mind is a programme. And information processing is more broadly defined than by universal Turing computation. Before digital there was analog computation or a start. Learn your history and stop making a fool of yourself.

And by admitting that when it comes to brains, NN's are at best an analogy, you are conceding my point. Brains aren't NNs. You just agreed that they're only analogies to NNs. — fishfry

It's not an admission. It's my point. NNs are successful models of the brain's essential functional architecture.

You talked vaguely of "biochemical processes". Well science prefers to talk precisely. And it seeks to understand the basic trick of the brain in terms of some replicable informational architecture.

But hey, I've lost interest. If all there is to do here is to keep correcting your misrepresentation of my arguments, that is really a waste of time.

The technical definition requires the physical manipulation of an infinite tape. — apokrisis

You mean an unbounded tape. Your ignorance is showing. Again.

If all there is to do here is to keep correcting your misrepresentation of my arguments, that is really a waste of time. — apokrisis

No amount of erudition can compensate for your bad manners. And as in our previous convo a few months ago, the more wrong you are on the facts [which several people noted at the time] the more condescending you act. Transparent and shallow, you fool no one but yourself.

And because we know intelligibility to exist, then we know that - whatever else -its unintelligible ground had to contain intelligibility as its potential. So we can actually know something usefully definite about fundamental unintelligibility.

This is apophatic reasoning. But hey, in metaphysics that is unreasonably effective. ;) — apokrisis

I don't disagree with anything here. We apprehend and comprehend the idea of finite, temporal being, and understand that it cannot be its own ground, which leads us to the idea of in-finite, eternal being. So, my question is, just as with finite temporal being we extrapolate to in-finite, eternal being; then why not from finite, temporal creativity to in-finite, eternal creativity, from finite, temporal intelligence to in-finite eternal intelligence, from finite, temporal order to in-finite, eternal order, and so on?

In any case whatever we think about that in-finite, eternal being is always going to be tendentious, always going to reflect our own prejudices; if we are moved to take any position as regards its 'attributes', it can only be on the basis of imagination mediated by logic; it can never be an empirical matter.

So, my question is, just as with finite temporal being we extrapolate to in-finite, eternal being; then why not from finite, temporal creativity to in-finite, eternal creativity, from finite, temporal intelligence to in-finite eternal intelligence, from finite, temporal order to in-finite, eternal order, and so on? — Janus

The extrapolation has to reverse a dichotomistic separation. It has to unbreak a symmetry breaking to recover the original symmetry. So there is a particular logical model to be followed.

So for instance, if existence depends on the actualised contrast between flux and stasis, or chance and necesssity, or discrete and continuous, or matter and form, etc, etc, then that definitely present distinction is what has to be folded back into itself as we wind back the clock to any vague and undivided initial conditions.

It is the unity of opposites argument. Dialectics. And I don't see that you are posing your own account this way.

The infinite is opposed to the infinitesimal. They represent the limits on the dichotomy represented by the notions of the ultimately continuous and the ultimately discrete. So a symmetric initial conditions would fold this distinction back into itself. It would be a state that is neither infinite nor infinitesimal. Neither continuous nor discrete.

That of course sounds rather mystical. But it actually maps pretty well to the Planck scale which was the start of the Big Bang. The geometric extent of spacetime was infinitesimal - as little as it could possibly be. While the energetic content of spacetime was infinite - as hot or dense as it could be.

So there is a logical formula to follow here.

You, on the other hand, are imagining a linear extrapolation. You start with some limited amount of something and multiply it until it grows to be unbounded. Time, creativity, intelligence, order and being are all finite and definite properties, so why can't they be - individually - infinite?

So nothing is being folded back into itself to heal a symmetry-breaking. There is no dissolving of the crisply divided to arrive back at a shared primal origin. The metaphysical operation you have in mind is instead turning a limited substance into an unbounded substance.

Instead of dissolving hylomorphic being by folding form and matter back into themselves via a loss of all distinctions, as in the notion of an Apeiron which is just pure fluctuation, you are accepting the substantial state and extending it without limit. It loses its located particularity by being rendered absolutely general rather than by being dissolved back to a vagueness.

The Peircean model is firstness => secondness => thirdness. Or vagueness => particularity => generality.

So sure, you can make absolute generality your initial conditions rather than your final outcome. But that then rules out a developmental logic.

You can see this tension playing out in theistic attempts to imagine divine immanence. Is God there at the beginning or realised at the end? Is God the creating intelligence who decided to construct a Cosmos for some reason, or is the Cosmos, through its evolution, the eventual realisation of Godhood?

Peirce's metaphysics argues that the development of the Comos represents the universal growth of reasonableness. The beginning was an unintelligible chaos - meaningless tychism. But that couldn't help but develop patterns and order. Habits emerged. The Cosmos started to self-organise and become intelligible. So the Cosmos is on a journey towards maximal "reasonableness". To the degree you want to read divinity into the story, the "designing intelligence" is simply the semiotic machinery - the fact of habit-taking - by which chaos can become completely ordered in a general or global fashion.

So this would be divine creation or divine intelligence of the most limited kind. Especially right back at the beginning. And even at the end, it only manifests as some general state of order. It is not intelligence as we mean it - a mind cracking problems for self-interested reasons. It is simply a mechanism - semiosis-driven self-organisation - extrapolated to the most global possible scale of being.

Again, modern science confirms this particular metaphysics. The Big Bang is self-organising its way to its Heat Death. The Planck scale symmetry breaking will become eventually as broken apart as it can physically be. It will arrive at the stasis of an anti-de Sitter void, a fully thermalised and unchanging dead universe.

Creativity and intelligence and mindfulness as we mean it are just passing negentropic eddies in this general flow towards a maximum entropy condition. We are not what creation is about. Even if we exist only by contributing to that entropification project.

So to extrapolate from us is metaphysically unjustified. At least if we are following a metaphysics that is based on the logic of dialectics.

And given that metaphysical dialectics proves itself to work, we should be brave enough to follow it all the way to talk about the beginning and end of substantial existence itself.

It is fine that you make your extrapolation argument that starts with the particular and abstracts to the general. But an unbounded amount of some stuff - like time, intelligence, creativity, whatever - is not actually a shedding or dissolving of boundaries. It is only a generalisation that leaves you with an unlimited quantity of that very stuff. The stuff is still bounded, still substantial and definite, even if you are imagining it to be actualised in some infinite quantity.

That's the problem. Stuff is hylomorphic. Definiteness is defined by the existence of a dichotomy - the unity of a complementary pair of bounds or limits. Your extrapolation only multiplies the quantities. It cannot dissolve the actual qualities in question. And so talking about an infinite amount of something fundamental solves nothing, just multiplies your causal difficulties.

A finite amount of intelligence or creativity is easier to explain than an infinite amount. And at least a finite amount, if multiplied enough, could become an infinite amount - using our metaphysical maths.

But to solve the problem of creation, the question of how existence could bootstrap into being, you need to be able to dissolve substantiality itself. You must undo the very notion of a quality.

And dichotomies - in defining a reciprocal or inverse relation - can do that. If you multiply x/1 by 1/x, you get 1. You can unmake your perfect asymmetry and recover a perfect symmetry. So now the multiplication goes in the right direction. The particular become not the general but the vague. You can no longer tell one pole of being from the other. Their particular qualities have been merged - mutually annihilated - to become again a featureless one-ness of unity of opposites.

So we have here two views that can be defined as mathematical operations. The metaphysical claims can be made highly precise.

The question then is which one is actually doing the trick? And which metaphysicians have talked about the opposed alternatives the most clearly?

I don't have time to respond in detail; but I will just point out that by 'in-finite' I don't mean to refer to "an infinite amount"; all amounts are finite. Likewise 'eternal' does not mean 'for an infinite amount of time" but 'timeless'; outside of the context of time altogether. The "virtual quantum soup" must be thought as both in-finite and eternal (not spatio-temporal, in other words) according to these definitions. It is not non-being in the sense of absolute nothingness (which is an absurd notion) but infinite, eternal being. If it "contains the potential" for intelligence and intelligibility, then it would seem to make more sense to think of it as in-finitely and eternally intelligent, than to think of it as brutely blind.

but I will just point out that by 'in-finite' I don't mean to refer to "an infinite amount"; all amounts are finite. — Janus

Yes. You sense the difficulty and try to avoid it.

I seek to make the difficulty plain and so force a definite choice.

If it "contains the potential" for intelligence and intelligibility, then it would seem to make more sense to think of it as in-finitely and eternally intelligent, than to think of it as brutely blind. — Janus

But here you have to offer the dichotomy on which your notion of intelligence, or intelligibility, or creativity, etc, is based.

You have to show that you are un-breaking a breaking rather than extrapolating a quantity to arrive at an "unbounded" quality.

My view of the development of human intelligence and creativity is scientific. It has that tested evidential support. And so I don't think of these named qualities as being in anyway physically fundamental or general. There is nothing more particular and emergent in the known Universe than the complexity of a living human nervous system.

So the dichotomy, the formal contrast, is between complexity and simplicity, between negentropy and entropy, between organismic level self-interest and purpose and physical level disinterest and blind tendency.

Peirce then connects the simple and the complex in psychological or phenomenological language. Intelligence (or any evolutionary/adaptive process) needs to combine selection pressure and spontaneously arising variety. Hence we arrive at the story where firstness equates to absolute blind spontaneity or tychism, and thirdness equates to absolute firm habit, the continuity of evolved constraints or synechism.

So now nature can be "intelligent" in that it has this evolutionary logic, this intelligible structure. A fruitful marriage of chance and necessity, freedom and constraint.

Human-style intelligence and self-centred purposefulness falls out of the picture. It is a general possibility taken to a particular extremity. To talk of a still "higher" creating intelligence has to be a continuation of that particularisation. A super-mind would have to inhabit a super-body as well.

Now we can imagine such a next step. The idea of artificial intelligence and the Singularity is one such extrapolation. Humanity could get downloaded to a technology that spreads itself across inter-galactic space.

A nice conceit. But it does correctly extrapolate whatever it is that we could mean by human intelligence and creativity as qualities that might increase in general quantity. If life and mind is negentropy that depends on accelerating the Universe's entropification, then spreading ourselves across the Universe to tap its physical resources at every possible location is what natural philosophy would predict.

But anyway, this is the issue. You have to choose whether you are magnifying a quality or dissolving a quality. And how can you head back to the origins of a quality by simply increasing the amount of it?

You claim you are not increasing the amount in trying to generalise the quality. But really, you are. You are imagining a little bit of local stuff spreading to take over everything. And that is simply shifting reality in the direction of one pole of some dichotomy. You are arguing that eventually - go far enough - and you lose sight of the other pole.

So take embodied human intelligence and creativity. You want to lose the necessity of the body and imagine the mind spread generally.

The Peircean view is pragmatic - mind arises as a way to regulate material physics, accelerate entropic flows. Mind makes no sense, it can't exist, unless it has that physical context.

So you are imagining a nonsense - a mind without that "other" which is the source, the cause, of its being.

I can see how tempting this move is. We are so used to thinking in terms of dualism. It is simply believed that mind and world are already separate, so both are free to grow in-finitely in their own realms.

But that is a dualistic metaphysics. And it doesn't in the end work. We know that. Hence even theists do try to find a more organic or immanently self-organising story occasionally. And Peirce spells out the logic of that.

So take embodied human intelligence and creativity. You want to lose the necessity of the body and imagine the mind spread generally. — apokrisis

In-finite mind is not "spread" anymore than infinite being is. You need to free your thinking from it's customary presuppositions to get this.

But that is a dualistic metaphysics. And it doesn't in the end work. — apokrisis

No, I'm not proposing any kind of dualism; that it might seem so is again due to your own prejudice. How can you tell, beyond its failure to gell with your own particular set of presuppositions, that a metaphysics is not working?

In-finite mind is not "spread" anymore than infinite being is. You need to free your thinking from it's customary presuppositions to get this. — Janus

Or you could explain what you are thinking as the alternative.

What even is in-finite being? You could be agreeing that it is the Apeiron - a potential not yet limited and thus not yet actualised. Why not explain in your own words to make it clear.

And what then is in-finite mind? If we follow the same formula, it is a potential that is not yet intelligibly structured and so not yet "intelligent".

Or maybe you are talking about sentience, or qualia, or soul, or something similarly substantive - a universal simple. So now you are either defending dualism or idealism. That becomes un-Peircean as Peirce is all about the psychological structure, the growth of universal reasonableness.

The Peircean claim is clearly argued in terms of an actual logical mechanics. The process is laid out in plain view.

Can you do something similar for "in-finite mind" if it is really universal sentience you are talking about, or universal will?

You talked about intelligence and creativity. That fits with a basically physicalist approach like Peirce takes. It avoids just presuming a dualism or idealism as the ontology.

But if you really meant in-finite sentience or in-finite will, that is what you need to defend.

No, I'm not proposing any kind of dualism; that it might seem so is again due to your own prejudice. How can you tell, beyond its failure to gell with your own particular set of presuppositions, that a metaphysics is not working? — Janus

Just because I tell you plainly what my position is, and then ask you to be plain about yours, doesn't mean I can't get past my beliefs.

It means I have arrived at my beliefs through a contest of the alternative views. And one of the fundamentals of epistemology is that offering up theories that are "not even wrong" is worse than a concrete theory that just is wrong.

So again, what I am hoping is that you will put forward a sharper account of the world-creating mechanism you have in mind as an alternative here.

If you are not talking dualistically, then what is this "mind" of which you speak? Is it synonymous with being, and so monistic in the idealist sense, or what?

If you leave me guessing, you can't really complain if I fill in your side of the debate too.

The problem is that an infinite pure potential that is not actual makes absolutely no sense. Remember we are both doing no more than trying to speak suggestively, apophatically, here about 'something' about which really nothing can be sensibly said.

The problem is that an infinite pure potential that is not actual makes absolutely no sense. — Janus

That doesn't help me understand what you could mean by in-finite mind here.

And I already said that - apophatically - the unbounded initial potential would be defined in terms of it being "not actual".

It's the same way we talk about "nothingness" - the absolute absence of things. But with nothingness of course, there isn't then a potential. Potentiality is what has been absolutely suppressed. So the difference with an Apeiron, Firstness or Vagueness is that we know it must have had a potentiality that was un-actualised. We know there is that actuality. So if we read it correctly - in terms of symmetry-breaking - that justifies our saying something concrete about that which is not the least concrete.

Again, I believe I have spelt out a metaphysics with an actual logical machinery. It is even mathematical in being framed in terms of reciprocal or inverse relations. It is certainly scientifically inspired in being a tale based on fundamental symmetry breaking.

So that is why I ask you to offer something as well developed if you want to argue for "in-finite mind". It seems fair enough to me.

I'll try to address this more later when I have time. For now I'll just say that the idea of an infinite potential that is not actual makes no sense to me. I mean what is the opposite of an actual potential? An imaginary potential, or a conceptualized potential?

I mean what is the opposite of an actual potential? — Janus

Saying a potential was actual is the after the fact view. So it says something was possible rather than impossible.

An infinite potential would then make anything and everything possible. At least at the "beginning". What is then now the actually possible vs the actually impossible is whatever actually happened and whatever actually didn't.

The reason some potential things would be impossible would be because constraints emerged to limit their being. Their existence was suppressed. And thus actuality gets defined by whatever it is that constraints can't suppress. Actuality is an expression of what can freely happen.

So you are wanting to make the initial potential some kind of concrete actual. You want it to be already substantial - limited by form so that it has definite being ... ahead of there in fact being any definite being.

But that is the mental hurdle you need to move past here. And I agree it is really difficult.

An infinite potential would then make anything and everything possible. At least at the "beginning". — apokrisis

That's the problem; I find it impossible to think that absolutely anything at all would have been possible even 'at', or 'prior to' the 'beginning'.

A universe of cartoon characters? A universe which is just a giant hamburger? A universe consisting of fairy floss? A universe where the inhabitants are heavier than the planets they inhabit? An infinitely complex and changing world which nonetheless consisted in absolute thermodynamic equilibrium? Or could any world such as our present one simply pop into existence 'fully formed' and without a history? I mean imagination's the limit; I believe I could think of potentially an infinite number of scenarios which would simply seem, not logically, but inherently physically, impossible, period.

So it seems impossible for me to imagine that there would not be an actual lawfulness inherent in the primordial indeterminate potential, that always already limits what could possibly come to exist. Of course, it's all speculation; but what tools do we have to work with other than imagination, intuition and the logic of what seems possible? I mean, we are endeavoring to determine what we should think about extremely abstruse matters, after all. And I do think it should, and probably inevitably will, remain ultimately an individual matter. We are not constrained by what the "community of enquirers" will ultimately come to think, because we cannot have any idea what that will be, or whether there will ever in fact be any overriding consensus.

A universe of cartoon characters? A universe which is just a giant hamburger? A universe consisting of fairy floss? A universe where the inhabitants are heavier than the planets they inhabit? An infinitely complex and changing world which nonetheless consisted in absolute thermodynamic equilibrium? Or could any world such as our present one simply pop into existence 'fully formed' and without a history? I mean imagination's the limit; — Janus

Have you already abandoned apophatic reasoning?

We start with where we are. We accept that there is a creation issue because we have the clear evidence that our existence has developed. Cosmology and fundamental physics then let us wind back our story all the way down to a quantum-level scale where radical indeterminism is known to set in - the Big Bang scale of about 10^-33 cm, 10^-44 secs and 10^32 degrees.

So it is from that set of physical conditions that the metaphysics continues to extrapolate.

Yes, imagination is needed. But it is now extremely constrained by the facts we are sure of. So all your suggested worlds, chosen because they are silly and contradictory, are already ruled out - unless apophatic reasoning finds some way they are a logical consequence of the "whatever" which would be the kind of potential which also produces such a highly constrained Cosmos such as the one that works to produce us.

So the question about other possible worlds would be about physical basics such as the number of dimensions, the strengths of constants, the number of different emergent forces. And none of your imagined universes suggested something different about those.

So it seems impossible for me to imagine that there would not be an actual lawfulness inherent in the primordial indeterminate potential, that always already limits what could possibly come to exist. — Janus

Well it helps to start by getting down to a starting point based on what we know. So it we are looking for what lies immediately before the Big Bang, we know that we are talking about some kind of quantum mechanics that lacks the kind of dimensionality which gives quantum fluctuations a strength and a direction in "our" universe.

And there is a ton of speculative physics on the issue if you want concrete proposals. There are models like the many loop quantum gravity approaches that seek to show how our 4D spacetime world could arise emergently from naked fluctuation. There are thousands of papers on the issue. There are computer simulations of self-organising spacetime metrics. People are trying to bring the right mathematics to bear on the question.

And I do think it should, and probably inevitably will, remain ultimately an individual matter. We are not constrained by what the "community of enquirers" will ultimately come to think, because we cannot have any idea what that will be. — Janus

Again, the scientific community is on to it. It's not about personal belief. It is going to be about whatever mathematical-strength model shows our particular dimensional set-up was always necessarily emergent from whatever an utter quantum indeterminacy can be understood to be.

There's a well defined approach and goal here. It's actually a pretty interesting story unfolding before our eyes if you check out the science.

Might be a good idea to start with trying to decide - to define - what a number is. — tim wood

A number can be many things. It can be careful symbol that we hold on memory. It can be a multiplicity that we hold in memory of something we observed or possibly created in our memory. They all exist as memory but not necessarily shared in other memories. I prefer thinking in terms of memory that is private and memory that is shared since everything ultimately is memory of our mind of some sort.

So all your suggested worlds, chosen because they are silly and contradictory, are already ruled out - unless apophatic reasoning finds some way they are a logical consequence of the "whatever" which would be the kind of potential which also produces such a highly constrained Cosmos such as the one that works to produce us. — apokrisis

So, the ways things are in our world that has emerged constrains how we can think about the the primordial, in-finite, eternal, virtual 'anything is possible'? We know what it is not and it is not anything we know?

Youo recommend checking out the current developments in science and I do find the idea of systems science and semiotics based metaphysics intriguing, but I am weak on math, and short on time, so I not confident I am able to adequately judge the validity and soundness of the kind of work you are recommending, since the validity and soundness is founded on the math. Also, I have always been drawn to the free flights of imaginative and intuitive thought, simply because I find I can do it, and I have a strong sense of the numinous which I am disinclined to give up on account of a belief that to do so would be to impoverish my life.

You have often accused me of being committed to a belief in a creator god, but that is not really accurate. I don't have any clearly defined belief about the ultimate nature of reality, and any religious feelings I might have had have not been strong enough and/or of the appropriate kind to have made me a church-goer or adherent of any particular religion.

I asked you before what would be a universal presuppositon-less criteria for judging whether a metaphysics "works". Is such a thing possible? I tend to think that the metaphysical ideas that are the most exciting and inspiring and able to elaborated into the most comprehensive systems are the ones that work best, since metaphysical ideas cannot be judged in terms of practicality, or reliably inter-subjectively corroborated.

At the moment I'm exploring Whitehead's works to try to gain an understanding of the overall movement of his thought. Some parts of The Concept of Nature (which I'm currently engaging with) are couched in mathematics, and I have a terrible time trying to prevent a loss of enthusiasm and interest when confronted with mathematics.

I have a strong sense of the numinous which I am disinclined to give up on account of a belief that to do so would be to impoverish my life. — Janus

That's fine. You seem interested and serious. I am just arguing for a particular point of view which represents a logical metaphysical methodology.

I asked you before what would be a universal presuppositon-less criteria for judging whether a metaphysics "works" — Janus

Any reasoned position must start from suppositions. How else could it work? The alternative would be perhaps some claim about "direct perception" - personal revelation - in regard to the truth of the Cosmos.

So yes, there must be some well-chosen suppositions to get the metaphysical game going. As I've said, the first is that something exists. The second is that something developed. The third is that development must be dialectical or dichotomous - a separating out or symmetry-breaking which speaks to a prior unbroken potentiality. The fourth is then that we must be dealing apophatically with some kind of "perfect potential" as the initial conditions.

Whether this metaphysics "works" as a whole then depends on how other alternatives stack up against it. It could either be challenged at some particular step (perhaps the Big Bang never happened, there was never a developmental story), or it could be challenged as a whole - as transcendent theism would likely do.

I have a terrible time trying to prevent a loss of enthusiasm and interest when confronted with mathematics. — Janus

I find Whitehead dire. Personally I would waste no time there.

The beauty of mathematics lies in its Platonic structures. It is the fact that fundamental abstract patterns can be the bones that underlie the flesh and blood materiality of the world.

A better read than Whitehead would be something like Ian Stewart's Why Beauty Is Truth: A History of Symmetry.

Your suppositions do make sense to and I dare say they would be essential to almost any conceivable consistent metaphysics.

I know what you mean about Whitehead, but I have been looking into him on and off over a period of perhaps 20 years and am gradually coming to comprehend the range and cohesion of his whole sweep of thought. So, I have found quite a deal of interest in his work.

Thanks for the Ian Stewart citation; I'll check it out. I have Nature's Numbers on my shelves. I've been meaning to read it for years.

I think you're confusing numerals with numbers.

As I said, numbers can take many forms. It is all how we envision them in memory.

Is number a one or a many? Numerals can be whatever you want them to be. Number - I don't think so. That's why the distinction between numbers and numerals. For example, if you look at a page from a telephone book you'll see no end of numerals, but not a single number. That's one reason a definition for number is good to have before talking about them. You're talking about how you think about them; I'm asking you to say what you think they are, in such term as might facilitate discussion, or at least guide it.

That's one reason a definition for number is good to have before talking about them. — tim wood

It's a philosophical curiosity that there is no definition of number in mathematics. In other words if you major in math, get a Ph.D. spend a career as a professional mathematician, you will never encounter a book or a paper that says, "A number is such and so."

A number is pretty much anything that's number-like, and the concept is historically contingent. First numbers were only the whole numbers and the ratios. Pythagoras discovered irrational numbers. Not till the middle ages did we start regarding zero and negative numbers as numbers. Not till Cantor did we regard transfinite quantities as numbers. There are all kinds of other mathematical objects we call numbers such as the p-adics, the dual and perplex numbers, nonstandard integers and reals, and so forth.

Yet there is not one single definition of number. It's an amorphous concept. Mathematicians "know one when they see one." I don't know if this has caught the attention of philosophers. But there is no definition of number.

Yet there is not one single definition of number. It's an amorphous concept. Mathematicians "know one when they see one." I don't know if this has caught the attention of philosophers. But there is no definition of number. — fishfry

Fair enough, but this only means that, as you say, there is no definition. That neither stops us - anyone - from offering one, or relieves us from the obligation to try, if we want to have any sort of reasonable discussion. It might start out, "For the purposes of this discussion, in which I intend to make points a, b, and c, I provisionally define number as...."

And I'll try one: number is that which has neither extension, substance, nor quality, but that expresses/represents quantity.

And I'll try one: number is that which has neither extension, substance, nor quality, but that expresses/represents quantity. — tim wood

Ok I'll play. Four questions.

* What is quantity?

* The imaginary unit i with i^2 = -1 ... what quantity does it represent?

* Do you regard i as a number?

* Does i exist?

Yet there is not one single definition of number. It's an amorphous concept. Mathematicians "know one when they see one." I don't know if this has caught the attention of philosophers. But there is no definition of number. — fishfry

Probably worth mentioning that category theory and structuralism have moved past this good old set theoretic view....

The theme of mathematical structuralism is that what matters to a mathematical theory is not the internal nature of its objects, such as its numbers, functions, sets, or points, but how those objects relate to each other. In a sense, the thesis is that mathematical objects (if there are such objects) simply have no intrinsic nature. The structuralist theme grew most notably from developments within mathematics toward the end of the nineteenth century and on through to the present, particularly, but not exclusively, in the program of providing a categorical foundation to mathematics.

Mathematical structuralism is similar, in some ways, to functionalist views in, for example, philosophy of mind. A functional definition is, in effect, a structural one, since it, too, focuses on relations that the defined items have to each other.

A structure is the abstract form of a system, which ignores or abstracts away from any features of the objects that do not bear on the relations. So, the natural number structure is the form common to all of the natural number systems. And this structure is the subject matter of arithmetic.

http://www.iep.utm.edu/m-struct/

So if it looks like algebra - it can be added, subtracted, multiplied and (perhaps) divided - then its all numbers.

And I'll try one: number is that which has neither extension, substance, nor quality, but that expresses/represents quantity.
— tim wood

Ok I'll play. Four questions.
* What is quantity?
* The imaginary unit i with i^2 = -1 ... what quantity does it represent?
* Do you regard i as a number?
* Does i exist? — fishfry

"Is" can be a tricky word. You ask "what is quantity?" Quantity is the general name for an idea that is always particular, and that refers to anything that can be quantified. Now I think you are confused in that you think a definition somehow "is" what something is. Or you apparently think that the definition of number, or quantity, will tell you what these things are. This approach or understanding - actually utilization - gets a lot of the world's work done, but it isn't remotely true. A definition is simple an agreed description, for some purpose.

As to i, it's the square root of -1, it's a number, and it exists (keeping in mind you probably have at best a partial idea of what "existence" means, and of what I mean by it). Definitions, then, are functional. And if any thing is going to be discussed in terms of its definition, or any understanding of what that something is, then it's best to start with some explicit expression of that definition or understanding. That's just good navigation. And of course it's negotiable, if that's appropriate.

It's a philosophical curiosity that there is no definition of number in mathematics. In other words if you major in math, get a Ph.D. spend a career as a professional mathematician, you will never encounter a book or a paper that says, "A number is such and so." — fishfry

Really? What is number theory about? What is Principia Mathematica about?

Sorry but you are bullshitting to an extraordinary level.

* What is quantity?

* The imaginary unit i with i^2 = -1 ... what quantity does it represent?

* Do you regard i as a number?

* Does i exist? — fishfry

You think numbers are defined in terms of quantity?

i represents the square root of -1.

i is a number.

i exists.

Simples.

In a sense, the thesis is that mathematical objects (if there are such objects) simply have no intrinsic nature.

A structure is the abstract form of a system, which ignores or abstracts away from any features of the objects that do not bear on the relations.

Sure, these work, for the people who use them - nice and functional. And if enough people agree, then whatever the problem is evaporates in a cloud of a kind of agreed ignorance. But we can still construct the question, "What is number?" It may not be a useful question that bears on the use of and utility of numbers - and in that sense it's useless, but are we all satisfied that the question is meaningless?

Given what definitions are, I argue that for so long as alternative answers can be generated, then the question remains meaningful. Whether the answers are worth the candle is a whole other question.