Kit Fine
I’m writing an interview with Kit Fine for the next issue of The Philosophers’ Magazine. Fine works at the seriously abstract end of logic and metaphysics, on such things as vagueness, parts and wholes, modal logic, and the objects of mathematics. Before you pass out, let me reassure you: it’s possible to get a grip on it, even if you couldn’t find your biconditionals in the dark.
What kind of things are numbers? Do they really exist independently of us or do we invent or construct them? There’s something to be said for both possibilities. On the one hand, numbers seem to have a “reality” that merely imagined objects lack. If you close your eyes, you can sort of “see” numbers, intuit them– they’re presented to us in a certain way, and there’s nothing we can do about it, just as objects are presented to us in vision. On the other hand, mathematicians have an extraordinary freedom to invent or extend the number series, and just “make up” such things as negative numbers or irrational numbers as and when they need them.
You’d think those two possibilities would be it – either we apprehend numbers or invent them – but Fine argues for something else entirely. Here’s part of what he said, “Mathematical objects are not exactly of our own making, but we actually have to do something to get them there. There’s something out there which we prod, but there’s the prodding that’s also required.” That’s marginally spooky, but maybe also right. What do you think — intuited, invented, or prodded?
Math is supposed to be consistent. It’s probably prodded, because one can’t just make them up without a consistency and I think numbers aren’t independent mystical things.
Seems like he’s right on the money. Over the years, a whole lot of prodding has been done, but only because there is the bare fact that there are amounts of things. If you only ever had 1 apple, you might not need to come up with the symbol, “1″. But once you want to start trading multiple apples, using symbols to talk about how many apples you want to trade makes sense, and it represents something real and tangible. As I said, we have come a lot further than that, but numbers certainly represent real things.
I can’t wait to read the interview.
My own understanding in this matter is that are in the main two viewpoints. There is on the one hand Platonism or Realism about numbers such that they are abstract entities having existence which is independent of any mind although they can be apprehended by a mind/s. On the other hand is Conceptualism, which rejects the existence of numbers in space, or any other imagined, or abstract domain of existence. However the conceptualist holds that numbers have existence in time, but depend on minds alone for the duration of that existence, that is they are solely mind dependent, mind created.
Kit Fine’s sympathy seems to favour the Platonistic viewpoint he states “Mathematical objects are not exactly of our own making, but we actually have to do something to get them there. There’s something out there which we prod, but there’s the prodding that’s also required.” I cannot really think that this is, in the light of our existing knowledge of science which includes all aspects of psychology and neuroscience the best explanation. It surely is not the simplest nor is it testable, compatible, although I would not dismiss it out of hand. Actually it is not really an explanation, more of an hypothesis. The first questions are:- “something out there; well what and out where?” “We prod; well how exactly do we do that, what does that mean?”
I would hold that if numbers are to be meaningful they must be used adjectivally. If I just say “11” it conveys little or nothing although I suppose it is in that case a noun. I need to say, “11 men” for there to be at least a shred of meaning. If numbers are adjectival then according to the Platonist all other adjectives must somehow find their home in some sort of existence outside of the human mind. We do not normally think of adjectives as having embodiment in any location other than our own imaginations, as perhaps we think, of nouns when we say for instance, “the tree is out there in the field.”
If an argument be made that existence can be such that space as we understand it is not involved and/or time neither. then I cannot cannot envisage such a state.
An interesting point arises here I think when we consider mathematicians who devote their talents to Number Theory. This as I understand deals solely in the complexities of the relationships that groups of numbers have to themselves or other numbers or groups adjectives do not come into it.. Like chess it seems self contained. Unlike chess it can be applied to other fields, but it can exist in the pure state. Where do those numbers originate, where do the logical connectives which enable us to calculate, do logic, originate. Surely the simple explanation is that they are a product of human ingenuity alone.
We might ask where were Numbers in the Cretaceous period no humans were around? Were they (the Numbers), all lurking somewhere awaiting the arrival of humans to discover them? Surely this cannot be correct? Assume now that primitive humans now evolve and they have fingered hands. Might it not be possible that the necessity arises for a person wanting to differentiate between one finger, two fingers, or three etc. accordingly refers to these different groups by some sort of verbal utterance? Thus numbers come into existence, a purely human concept.
James Garvey states “If you close your eyes, you can sort of “see” numbers, intuit them– they’re presented to us in a certain way, and there’s nothing we can do about it, just as objects are presented to us in vision.” That does not seem to happen for me. Maybe that’s where I am going wrong.
Don,
‘if numbers are to be meaningful they must be used adjectivally’
In ordinary language cardinal numerals may often be used ‘adjectivally’ (or rather as noun-modifiers) – “there are eleven men” (or “11 men”) but I am uncertain this tells us anything about numbers. Numerals can also be used substantively in ordinary talk and whilst some such usage can be easily translated from ‘substantive’ to ‘adjectival’ (e.g. ‘the number of men is eleven’) it seems others can not e.g. “11 is the prime pair of 13.” The latter seems meaningful enough (and presumably numerical symbols carry meaning in mathematical proofs too) and it does sound like we are talking about objects in such an instance – exactly the type of objects that those working in pure mathematics seem to make discoveries about (doubtless they will discover a new highest prime in due course for example). And with regard to the early human wanting to differentiate between two fingers or three who accordingly refers to these different groups by some sort of verbal utterance, really this seems like an explanation of how numerals came into existence – not numbers.
I really know nothing at all about the philosophy of mathematics but if the simple explanation is that numbers “are a product of human ingenuity alone” – are invented as opposed to discovered – we do seem to need to explain why eminent philosophers in that field are not satisfied by it.
Were there numbers in the Cretaceous? Fine argues that numbers necessarily exist, so they’ve always been there.
Re Curious July 22nd.
My argument here is to challenge the idea that there is some sort of state of existence which is not dependant on either or time or space or an idea in an animal’s brain or all three of these. The platonic Idea of Forms as I understand, posits the existence I do not know where, of perfections, from which things in the real world are but imperfect instances. This will include numbers. So If I am correct the problem of where numbers have their origin has basically two arguments. These are, as have already mentioned before are Platonism and Conceptualism. I favour Conceptualism, Which holds that numbers are mind dependant, mind created.
The best argument against Platonism I have seen is as follows:-
1 (Suppose Platonism is true. Then:
2 Numbers are not located in spacetime. Hence:
3 We cannot have any causal interactions with numbers. But:
4 In fact we do know lots of things about numbers (for example seven is less than nine) Therefore:
5 Platonism cannot be true. cf Michael Jubian, Contemporary Metaphysics p.27.
It occurs to me here that if the Forms of Number/s are perfection; it is difficult to see how we can calculate with numbers so exactly as we do if they are merely imperfect instances of the Forms.
I do not know exactly where Kit Fine would go to do his “prodding”, It could well be what I say here is irrelevant to ideas he may have.
Numbers are as it appears, not concrete entities in the sense that trees cats and houses are but that is not to say they have no existence. They exist in the same manner as say facts, values, ideas, proposals, rules, relationships and so on. These products of the human mind may not have spatial existence, (unless you are going to argue they reside at some location in the brain) but they do have temporal existence. I Suppose someone could dwell on the number seven for five minutes at the exclusion of all else. My contention is that numbers begin and end in the human brain.
Yes this subject is deep and complex and as you say eminent philosophers oppose each other thereto. Impossible to do full justice to it here or anywhere for that matter without much research and composition My interest was only spurred by the statement “Fine argues for something else entirely. Here’s part of what he said, “Mathematical objects are not exactly of our own making, but we actually have to do something to get them there. There’s something out there which we prod, but there’s the prodding that’s also required.” I thought this was highly hypothetical at best and could only think it maybe had something to do with the Platonic approach to the subject, I may well be barking up the wrong tree here. I dare say all will be revealed in due course in the next issue of TPM.
I find it of interest that this subject so far has attracted only five replies. Were it on Lesbians, Politics, or Ethics replies would flood in. Some hold that the only true philosophical subjects are Metaphysics and Epistemology.
Re James Garvey.
Were there numbers in the Cretaceous? Fine argues that numbers necessarily exist, so they’ve always been there.
At the risk of appearing completely idiotic I must point out we have unearthed millions of fossils from the Cretaceous period but never a fossil number. So where exactly are these numbers? Obviously they are presently in a type of existence which transcends fossil (material) existence. I dare say, that being the case we somehow, if we agree with Kit Fine, become aware of them. This does not seem to me at this moment to be the best explanation. It is certainly so far as I can see not testable nor is it compatible with present scientific thought, and it is certainly not a simple explanation. As an argument concerning the problem of numbers it is certainly relevant, and I suppose it has predictability concerning what other stuff is similarly lurking out there, quite a lot I imagine. One thing seems certain that without a human brain, numbers remain undiscovered. We know quite a lot about the brain currently. Maybe some how we will eventually find exactly how the brain comes into possession of numbers either by its own processes or additionally by something else which at the moment for me at least seems to have doubtful ‘existence’. I look forward to the next TPM and possibly being won over by the argument.
James,
If numbers necessarily exist does that not make it rather misleading to say “they’ve always been there” or indeed “there were numbers in the Cretaceous”? Aren’t numbers ‘outside’ of time i.e. eternal rather than sempiternal (existing for all time) – or is that part of the novelty of Fine’s argument?
Hi Don,
Thanks for the reply. I appreciate that this subject is sufficiently deep and complex that it would be impossible for you to do it justice here. That you have the wit to usefully respond to these gnomic remarks on such a deep subject without much research and composition is very much to your credit.
The conceptualism (psychologism?) you are inclined to adopt with regard to numbers maintains that numbers do exist but (only) as concepts in the mind. And given very plausible physicalist assumptions, you say that numbers ‘begin and end in the human brain’ and have, if not spatial, at least temporal existence. It makes perfect sense to ask whether the concept of a given number had emerged by a certain date but we seem to run into difficulties if we cannot distinguish between the mind-dependent concept of a number and the number itself. The caveman with no concept of number has no number of fingers because numbers do not yet exist and if homo sapiens become extinct “seven is less than nine” will cease to be true. Large numbers spring into existence when people count high enough and an infinite number of possible numbers must remain non-existent because there will not be an infinite number of number ideas (and sentences about all natural numbers will fail to be true on this account).
As Frege put it: “Weird and wonderful…are the results of taking seriously the suggestion that number is an idea” and as I understand it conceptualism with regard to mathematical objects has very few adherents. Quoting from the Stanford Encyclopaedia of Philosophy (to which I have been referring) “you might doubt that there really is such a thing as the number 3, objectively and independently of us, but you should not for that reason claim that your idea of 3 is 3, for that is just a confusion — it is like saying that your idea of Pegasus is Pegasus…” I wonder if in fact nominalism might be nearer your intuitions. As the SEP puts it (in its entry on ‘Platonism in Metaphysics’) “a nominalist about numbers would say that while there are such things as piles of three stones, and perhaps “3-ideas” existing in people’s heads, there is no such thing as the number 3” – this seems in tune somehow with what I think you may have been trying to get at with your comments about the adjectival use of numbers. And in any case if you are saying that numbers “begin and end in the human brain” it does sound rather like you are expressing disbelief in the existence of numbers – one can imagine an atheist saying of God: “He begins and ends in the human brain”.
I find that, without positing any sort of spooky information transfer contact between humans and abstract objects, the SEP entry does offer some possible responses to the epistemological argument against ‘mathematical platonism’ which (like nominalism) holds that if there are numbers they are abstract objects. One of these, the approach of ‘Plenitudinous Platonism or full-blooded platonism, (FBP) seemed particularly interesting. The jist is that all the mathematical objects that possibly could exist actually do exist, and that therefore every purely mathematical theory that could possibly be true (i.e., that’s internally consistent) accurately describes some collection of actually existing mathematical objects and thus in order to attain knowledge of abstract mathematical objects, all we have to do is come up with an internally consistent purely mathematical theory.
Whether this line of thinking can tie into Fine’s ‘prodding’ comments I really don’t know – by prodding mathematical theories to see if they are consistent we find out truths about the abstract realm of numbers perhaps?
In any case I do enjoy the blog postings that invite you to go find out how much you don’t know about something important. (Thanks James.)
When I think about mathematics and numbers, I think these are human made objects and tools that we have developed to help us understand real things that exist either inside or outside of ourselves. These real things can be objects, processes, relationships, etc. The numbers (symbols) are “real” because we have made them and we use them to discover other real things.
All interesting stuff. Sorry not to be able to take part more fully — deadlines … deadlines everywhere.
About numbers always being there, existing necessarily, etc, Fine did say something like, “if you want to speak this way”, we can think of numbers as always existing, but he might think that’s another way to say they’re not in time. It’s not clear, but that’s my fault.
The prodding business is Kantian. There’s something out there, but we have to do something to it to get numbers — just like something is given to us in experience, but we have to categorize it to get a world of objects.
Here’s one of Fine’s papers which might help
http://silverdialogues.fas.nyu.edu/docs/CP/299/fine.pdf
Calvin Johansson has written as follows
“When I think about mathematics and numbers, I think these are human made objects and tools that we have developed to help us understand real things that exist either inside or outside of ourselves. These real things can be objects, processes, relationships, etc. The numbers (symbols) are “real” because we have made them and we use them to discover other real things.”
The following is apparently what Kit Fine holds.
“Mathematical objects are not exactly of our own making, but we actually have to do something to get them there. There’s something out there which we prod, but there’s the prodding that’s also required.”
The statement by Johansson has to my mind far more mileage in it than that of Fine. There is surely no doubt that in the absence of a human brain the experience/existence of numbers is not possible. So the more fruitful path of enquiry seems initially to enquire into the structure of the brain so far as numbers and all that they embrace are concerned. Google is of course the first port of call here and appropriate articles can be found. Compare this with the suggestion that “something exists”, “out there”, “needs prodding”. Where does one go with that? It is something like inventing a God to explain all Human existence. A preliminary problem is invented wherein God has to be explained. This just sets the enquiry concerning existence back one large step. I guess that much work has to be done before we can conclude that numbers are a metaphysical entity and can only be considered philosophically.
I have posted this before reading Fine’s paper as advised by James Garvey so maybe enlightenment will shortly descend on me. I do not anticipate an easy read.
“The numbers (symbols) are “real” because we have made them and we use them to discover other real things.”
Calvin, it seems you equate numbers with digits – “numbers (symbols)” – so presumably you would say the same of language and letters as you would of mathematics and numbers – that they are “human made objects and tools” that “help us to understand real things that exist either inside or outside of ourselves.” (As I recall Wittgenstein compared words to tools in a toolbox – perhaps mathematical symbols are usefully thought of in this way?) Presumably what you explicitly say about numerical symbols you would also say of letters (or words) i.e. that they are ‘…“real”…’ (in quotation marks – scare quotes?) “because we made them” and can “use them to discover other real things”. Intuitively I don’t know that the utility of inventing tally marks (or their successors) shows that mathematical objects are invented but perhaps you are on to something about how the naturalist might explain (away) numbers.
Don, certainly there is something to be gained from inquiring into the structures of the human brain concerned with the ‘experience’ of number (although it does not seem to follow from the fact that x cannot be experienced without a suitably evolved brain that the nature of x is explained by looking at brains). If you wish to pursue that avenue of inquiry you need not confine yourself to human animals. No Clever Hans or Alex the parrot in sight – there are studies suggestive of the fact that counting skills are possessed by other species. Conceptualism might be badly flawed but that does not mean we cannot find an account that will satisfy the the ontologically parsimonious and the physicalist. Looking at our linguistic and counting skills may well be the way to dissolve what is really a puzzle. But it seems much ‘therapeutic’ philosophical work has to be done before we can conclude that numbers are not metaphysical entities – there is a great deal of doubt that numbers would not exist in the absence of a human brain. With Fine’s gnomic remarks (considered out of context) we could, of course, only have ever gone so far. But it appears we have gone far enough to bring out the serious problems with both Mathematical Platonism and Conceptualism. And that at least is useful in helping us understand why it is sensible that Fine “argues for something else entirely”. What it is that he does argue for I shall try and get some grasp of in due course – though I have little hope that I will be able to follow a Kantian turn on the philosophy of mathematics. Interestingly it had occurred to me earlier that it could be said there is no mathematical *object* in the absence of a subject (a faint echo of Schopenhauer). I had discounted this as a trivial remark at the time but if there is something right about the Kantian turn James suggests Fine has taken it might confirm your belief that there can be no mathematical objects without human brains.
Re Curious
Presently man has a human take on the world. This take has been developed such that predominately, survival of the human species is maintained. So presumably much which may be out there is irrelevant so far as survival is concerned. Kant believes that we can neither intuit nor have any determinate cognitions of the nature of objects in themselves, and that these (due to the transcendental ideality of our forms of intuition) “are not objects of our senses”. When I use the expression “Out there” I mean something akin to Kant’s Noumena.
Putting it roughly and remembering I am not a Kantian scholar, it seems to me that we live in the world of Phenomena as opposed to the world of Noumena, things in themselves. with which we have no direct access. Consider Colour and Sound; these are human experiences and not to be found outside of that. Thus there is no sound when the tree falls alone in the forest only longitudinal waves of air compressions and rarefactions. There is no colour in the deserted room only reflection of electromagnetic waves from certain substances which are selective in which waves are reflected and absorbed.
The point which I am rather laboriously trying to make is that care must be taken when we consider periods of time before the existence of intelligent life on Earth. To project our modern Human take back to those times can lead to misunderstandings. There was no Sound, Colour, Heat, etc just a continual process, no causes and effects, that is a human construct aiding human understanding of the environment. So to find numbers there or the human understanding of numbers is rather like expecting to find in addition to books, loaves of bread on the shelves of the university library.
Hello Don,
Thank you for your comments.
By “out there” in your comments of July 24, 1:01 pm I was, for some reason, under the impression you were still talking about something like the world of Plato’s forms rather than Kant’s noumena. In any case, your position continues to be that there can be no numbers prior to the existence of intelligent life on Earth. The problems that supposedly come with holding that position hardly need repeating. But as Fine notes (I am only just beginning to scan the paper James linked to) presuming there was a time when numbers first came into existence (perhaps ‘when man first began to count’) apparently ‘makes a mockery of how we wish to apply mathematics.’ And Fine must think that Frege’s objections to conceptualism/psychologism – as they apply to these and other ‘anti-platonist’ positions must be pretty hard to deal with in a way consistent with physicalism if it occurs to him to turn to Kant for a way out. Even he conceded until recently that “given the obscurity in Kant’s original doctrine of transcendental idealism, the prospects of making out a similar doctrine in the case of mathematics do not look good.” I imagine we would both prefer to explain (away) numbers naturalistically and we would both very much like to do that without Kant (“the wild Irishman of Teutonic thought” who recklessly galloped at and crashed through philosophical hurdles shouting “a priori!”). And I have been trying to think about naturalistic ways to do that – thus I did pick up on what looked promising to me about things said by you and Calvin. But I failed to see anything offered up against Frege’s criticisms.
There is a short paper titled ‘Numbers and Ideas’ by John Burgess that I have come across. In his account the true debate is not between Platonists and Conceptualists but Realists and Nominalists (who hold that numbers are useful fictions). Burgess (John N. Woodhull Professor of Philosophy at Princeton) wants to argue for Realism over nominalism but thinks what is most important is convincing people that ‘realists and nominalists are right in their common opposition to conceptualism’. Burgess says that “I have never known a single realist who was in any meaningful sense a Platonist. What is actually the case is that anti-nominalists take much more seriously than nominalists the thought that mathematics is a human creation, since mathematics is a body of theory expressed in language, and language is a human creation. Now creating a language involves creating certain rules for its use. Among these is, I believe, a rule to the effect that tense and date are not to be applied to mathematical existence assertions. One can say ‘There exist infinitely many prime numbers,’ but to ask ‘How many of them already existed in 1000 BCE, or during the Cenozoic Era?’ is to commit a kind of grammatical solecism. Nominalists say they are opposed to the view that numbers are ‘eternal’, existing ‘outside of time’. But to say that numbers are ‘eternal’ is a misleadingly Platonistic way of putting the simple negative grammatical fact of the inapplicability of tense distinctions in mathematical contexts. That simple grammatical point is all the realist really believes about the ‘timelessness’ of number.” I do hope the truth is that non-Kantian direction.
Returning, reluctantly, to Kant and the point you try so laboriously to make (I do appreciate your labours in the face of my continued misunderstanding), you warn that ‘care must be taken when we consider periods of time before the existence of intelligent life on Earth.’ Indeed so. Of course it seems there could very well be colour, sound heat (and causality?) ‘before the existence of intelligent life’ if there were suitable non-human non-intelligent subjects of experience. But of course there need not be such subjects and we would naturally want to say that if you go back far enough there won’t be any subjects of experience at all and so no colour or noise. And you would want to say that there are no numbers or understanding of numbers there. Given the first stage of Kant’s Transcendental Aesthetic there is also no time ‘before’ the existence of experiencing subjects – there simply is no ‘before’ the existence of experiencing subjects and hence no colourless and noiseless ‘continual process’. As well as a lack of numbers, I believe the noumenonal has no time or space for libraries for you to fail to find loaves in.
“The nominalists assume that they have an understanding of what it would be for a mathematical object or entity to exist that is independent of ordinary mathematical standards of sufficient proof, by reference to which understanding they can criticize the ordinary mathematical standards. So-called realism is really just skepticism about the existence of any understanding of what ‘existence’ means in mathematics that is independent of ordinary mathematical standards for evaluating existence proofs. The nominalist denies the existence of numbers, while the realist denies that the nominalist understands what is meant by ‘existence’ as applied to numbers. Thus the realists think the nominalists are confused. But realists and nominalists agree that the conceptualists are confused…”
http://www.richmond-philosophy.net/rjp/back_issues/rjp4_burgess.pdf
Re Curious.
Familiar with Russell’s paradox and how it wrecked Frege’s work at the eleventh hour and the example of the barber paradox I persevered with Fine’s paper sort of hanging in there but with some misgivings It cannot be that difficult, I am thinking. “Here are a couple of pictures that may help” says Fine, Great I think, but Bu**er all is there, I gave up. Let’s get back to Stangroom and Benson’s “Why Truth matters” Wherein incidentally is stated on P.16 “We simply allow ourselves, without much worry and reflection to assume that the way humans want the world to be is the way the world is, more or less by definition – and endemic confusion and muddle is the result.”
Whilst my academic training was in Philosophy I can never divorce myself form initially taking a scientific viewpoint and stance to problems which I believe have possibilities in that connection and the problem of numbers is one of them. Remember how Bergson’s Élan Vital, Vitalism, eventually capitulated to the accumulation of scientific knowledge. I am not suggesting that science will solve this one (numbers) but philosophy so far can do no more then put forth theory and counter theory. In this connection when I see words like Transcendent, Emergent, Epiphenomenon as part of explanations I begin to feel uncomfortable these words seem in themselves to have little explanatory power or value.
As I have said before and I will speak for myself. I believe that my take on the environment is predominately such that survival is predominant. The evolutionary process, which for me is the best explanation, has caused most animals to have an instinct of curiosity. Very highly developed in the human it has been turned to other things, which perhaps are not on the face of it about survival. That is disputable, but another matter. So far as I can make out all I really contemplate first hand are my own ideas or more simply my own brain processes. Thus I do not have immediate contact with, for want of a better term, what we might call Reality or what I might call my Noumenon as opposed to my Phenomenon. I believe that in the final analysis Concepts like Scientific realism which posits entities like electrons and quarks do not describe the world as it may be but of course pragmatically it all seems to work. As Niels Bohr said “It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about Nature”. So that leaves me in a situation as a human being in terms of a human to endeavour to merely say things about Nature. I am not adding much to the main argument here other than to try to give some indication of my responses to this place in which I find myself for the last goodness knows how many years. James Garvey stated “If you close your eyes, you can sort of “see” numbers, intuit them– they’re presented to us in a certain way, and there’s nothing we can do about it, just as objects are presented to us in vision.” This just does not happen to me. Numbers hold, outside of philosophical debate, no mystery for me.
Fine states in his Paper “Mathematical Existence” that the solution to this problem may well lie in an intermediate position. Without going into details this reminds me of the Compatibalist viewpoint in the problem of Free will v Determinism, it a bit like a lame excuse. In fact science currently has much to say in opposition to the concept of free will.
So what more can be said about Numbers. One conceptualist suggestion is for him to concede that numbers are not ideas having their origin in the mind, but are in fact properties of things, ideas. Out of this it can be argued that different people having the idea of seven all have a certain property in common and it is this property that actually is the number seven.
Another explanation embraces the idea of a group which would make the noun say three, redundant. We could say there is a three membered group of men in a room. The phrase “being three membered” does not include an occurrence of the word three as a noun,. There is not time to develop these kinds of theme here, Personally I would prefer to pursue the problem neurologically. If there is a solution this does seem the best path to follow.
There will, of course, be matters of fact regarding when triangles were first thought of, talked about or represented. But it occurs to me that it is confused to say triangles existed before life evolved on Earth and it is just as confused to make the contrary claim that they did not exist until man had reached a certain stage of evolutionary development. This is not, I think, because triangles occupy some ‘timeless’ Platonic realm, it is just because if you are in that dilemma – were triangles discovered or invented? – you have got caught in the fly bottle. I’m becoming inclined to think that the discovery/invention question about mathematical objects is not a problem to be solved but a confusion to be dissolved.
Re Curious:-
“I’m becoming inclined to think that the discovery/invention question about mathematical objects is not a problem to be solved but a confusion to be dissolved.”
If not that, then maybe as John Dewey said, “Old philosophical problems were never resolved they simply stopped mattering”
I still cannot help thinking that numbers are initially, in the nature of labels which we prepare and attach to certain entities in order to compare their relative sizes. Out of this seven is less than nine because WE do not permit it to be anything else.
“The question is,” said Alice, “whether you can make words mean so many different things.”
“The question is,” said Humpty Dumpty, “which is to be master that’s all.”
Having been alluding to Wittgenstein, I supposed it might be worth trying to see what he had to say about mathematics. Rather a lot it seems, looking at the SEP entry on “Wittgenstein’s Philosophy of Mathematics.”
One of the things Wittgenstein argued – in his intermediate period of ‘constructive formalism’ – was that (according to the SEP) “mathematics is essentially syntactical, devoid of reference and semantics. The most obvious aspect of this view, which has been noted by numerous commentators … is that, contra Platonism, the signs and propositions of a mathematical calculus do not refer to anything”. Wittgenstein says: “[n]umbers are not represented by proxies; numbers are there.” Not only are numbers there in the use, numerals are the numbers, Wittgenstein: “[a]rithmetic doesn’t talk about numbers, it works with numbers.” I found this interesting because one of the things I had felt was a source of confusion in earlier comments by yourself and Calvin was a failure to clearly distinguish between digits/numerals and the numbers those symbols referred to. So it seems I wasn’t questioning deeply enough – I simply couldn’t help but think that numerals must refer to numbers.
According to Wittgenstein’s formalist views, in doing mathematics we operate in a purely formal, syntactical manner. and this ties in with the view that “everything in mathematics is invented” that is constant in his middle and late philosophy. The ‘middle’ Wittgenstein says “[w]e make mathematics,” and the later Wittgenstein says “the mathematician is not a discoverer: he is an inventor.” But wittgenstein does not only reject Platonism – he also rejects the ‘standard’ view that man invents mathematical calculi, but thereafter discovers a finite number of its infinitely many provable theorems. Unusually, it seems, Wittgenstein rejects the view that we discover truths about a mathematical calculus that come into existence the moment we invent the calculus. Getting clear on that ‘standard view’ was helpful for me. And I wonder how this ‘standard view’ connects up to Fine’s ‘procedural postulationism’ – this being the view that mathematical objects are the product of postulation but what is postulated are procedures for the construction of a mathematical domain not propositions and indeed his talk of transcendental idealism.
Anyway, I must say I find all of this deeply puzzling, if its a fly bottle I’m stuck there is a lot in it to explore.
If you take the Kantian position (or at least his position as I understand it) you arrive at a much neater picture altogether.
There is a noumenal reality, there are the processes of categorisation, and there is the phenomenal reality that results.
Mathematics lies entirely within the categorisation processes.
The confusion arises because the deadly lure of Realism, that the phenomenal world is all there is, and that consciousness is an emergent property of it, is the accepted way science does what science does.
If you accept – even as an ad hoc proposition – that the phenomenal world is an emergent cross product of consciousness and some other noumenal reality, then ‘what numbers are’ and ‘where they are’ becomes simple.
They are algorithms – Kantian categories – used by consciousness to organise a noumenal reality onto a phenomenal one. And since space and time are real only in the context of such organisation, it makes no sense to ask if numbers existed before people thought of them. The past does not exist except as a linear projection backwards from a never ending present, whose sole justification is that we have memory of ‘times past’ .. Time and mathematics are just ways of looking at things that are so intrinsic, we have forgotten that we made them up..
Numbers are not ‘Kantian Categories’. And the Categories are not usefully described as ‘algorithms’ – they are not problem-solving procedures that we follow but that which allow us to be the type of things that follows problem-solving procedures.
The phenomenal world is not ‘an emergent cross product’ of consciousness and some other noumenal ‘reality’. The word ‘emergent serves no purpose here, and the phenomenal is not in any way a ‘product’ of the noumenal ‘reality’ because being a product of something implies causation and the noumenal does not ‘cause’ anything. As for ‘consciousness’, if that is meant to refer something which is itself experienced then it does not use Categories but has Categories applied to it and is part of the phenomenal world (not something that prodiuces it). If by ‘consciousnes’ you mean to refer to the unknowable “I” then you refer to a thing-in-itself – an item of the noumenal. Anbd thus not something that works ‘with’ the noumenal to ‘produce’ anything else.
Time is not a Kantian Category, along with the other precondition of experience – space – it is a form of intuition. And Time and that part of mathematics that is a body of a priori synthetic knowledge – are not ‘just ways of looking at things’ and it is not the case that we ‘made them up’.
If I am confused I am sure you will tell me the ‘deadly lure’ which has caused it.
Curious:
Very simple. The deadly lure of Certainty which allows you to state equivocally that this element is not a member of that set. Etc.
All you are saying is that my model is denied by your model unequivocally.
I never doubted it. Now look at my model as a description, as a reordering of concepts into different sets.
As a thought experiment.
That’s all. If you change perspective, where things seem to belong changes. Then you may see more. If you persist with a single Certainty, then there is nothing more to be said
I maintain that whatever you care to call it, Time and Mathematics are indeed just a way of looking at things, and yes we did make them up. But the we who made them up is somewhat beyond the you that cant grasp that particular concept.
It occurs to me that many people study philosophy academically as a result of a deep need to arrive at certainty: You have to accept that I find it as quite the reverse, it makes the only certainty that there is no certainty.
A model is only a model, a limited description of something larger.
My proposition is that the world as we know it is just such a model, and the model is written in the language of mathematics, energy and spacetime, but that is not because it is so, merely because those are tools we somehow have access to.
I assume that was what Kant was trying to say. But with Kant, ‘Semper disputandum est’
‘All you are saying is that my model is denied by your model unequivocally.’
Er, no, all I was saying was that you are wrong about Kant on all the points I have enumerated – none of which you respond to. I have made no assessment of your ‘model’, you did not claim you had presented one of your own, and I did not realize I had presented one that ‘denied’ yours. It looked like you were simply making some claims about Kant (and showing some sympathy towards them), and I thought Iwas simply saying you were wrong about Kant with respect to x, y and z. (And there are matters of facts about some of these things, some of which I will no doubt be wrong about).
I have no certitude that your model is not, for some purposes, a good model. And it does seem useful that you distinguish your model and proposition from those of Kant. Presumably you feel there is no point trying to elucidate who the “we” that ‘made up’ Mathematics as ‘a way of looking at things’ are supposed to be if you feel they are ‘beyond the me who cannot grasp the concept’. But my empirical self is genuinely interested to know and, more to the point, I would also be very interested to know if ‘what numbers are’ and ‘where they are’ remain simple enough questions that you can actually answer them.
The table of Kants categories
Quantity
Unity
Plurality
Totality
Quality
Reality
Negation
Limitation
So far we have all the basic properties of all mathematical relationships..
Relation
Inherence and Subsistence (substance and accident)
Causality and Dependence (cause and effect)
Without a space time phenomenal world in which a phenomenon is a function of space-time matter and energy, there is no causality, no inherence and no subsistence…
Community (reciprocity)
Another essential mathematical concept.
Modality
Possibility
Existence
Necessity
Pretty much mathematical.
Perhaps you haven’t understood Kant?
He didn’t express it the way I do, but its the same picture. The same process.
Leo,
Thank you for the ‘cut and paste’ from Wikpeadia. But it does not answer the ‘simple’ questions or alter the fact that your post about Kant is riddled with errors – not all of which are mere matters of terminology.
I am certain of very little. But I am certain that I do not understand Kant. And I am just as certain that you do not either.
To say you have the same picture as Kant is plain silly. The man was a genius who constructed a massive philosophical system that that tried to solve – and certainly raised – deep philosophical problems. You have not even seriously looked at that picture, never mind understood it.
This is not to pretend that I have.
Oh well. It was as they always said, what you got from a Cambridge education.
The implicit assumption that you could understand things from first principles and have the sheer arrogance to announce what you had reasoned even if no one else in the world agreed with you.;-)
All I can say is I have spent 40 years pondering this from a completely different aspect, and what Kant seems to be saying as far as I can tell is broadly similar.
Maybe I am a genius too. Who knows?
I am certain of nothing at all. Except the certainty of absolute uncertainty.
My philosophy, if such it may be called, is to go back to first principles, at which point there is nothing. No certainty at all.
I merely stand here remarking that if you want a certainty, it will always be at some level, an act of faith. Which bumps the quest for knowledge up a level from ‘what can I be certain of’ to ‘how can I achieve functionality without certainty’. Since I personally like to function.
Mathematics simply uncovers in a pedestrian linear fashion the qualities we have put into the world. Of dimension, of interaction..once we have that worldview, mathematics is implicit in it already. We are merely making conscious that process by which we say ‘that is bigger than this’ and so on.
It’s all there right up to the purest maths there is.. the qualities of curviness, of boundary..of symmetry and rotation…of interactions.. that mathematics is intrinsic in our world-view. What we do in mathematics, is to assign quantitative and qualitative properties to things so we may communicate about these things. It is essentially a refined aspect of language.
My point is that the world-view is something we construct along those lines, therefore mathematics is ultimately not in the world produced by that world view, but in the way in which we construct it.
The confusion arises from denying the existence of a consciousness independent of perceived reality.
Once you accept that the very existence of a being in a reality implies some interactive boundary where perception of ‘self’ and ‘other’ forms, and call that ‘consciousness’ then there is a place for all these ‘tricky’ things like maths.
Now for me it all seems totally trivial, and there are no ‘tricky deep philosophical problems’ Or rather there are only about three.
Accepting a three-part rather than a dualistic model solves the lot, really.
There. That should be enough to enrage you with its sheer intellectual arrogance
My Dear Leo,
It is not intellectual arrogance that I detect. And I’m not at all enraged.
Never directly responding to criticisms, never admitting you are wrong, forever changing the subject or being otherwise evasive and continually conflating your ideas with those of others whilst insisting that you worked it all out by yourself when ‘it’ remains unsupported by good argument and ‘it’ is nothing new.
40 years of pondering? In the desert?
Re Curious:-
I am wondering if you have read Leo’s interesting paper found at http://www.shaman.co.uk/downloads/The%20Philosophy%20of%20Experience..pdf
This does give some insight into his own metaphysical conclusions on matters, which interest us all.
So far as Kant is concerned, for the beginner, or those who have read and forgotten, I recommend Nigel Warburton’s “Philosophy The Classics” Chap 10. I hasten to add am not suggesting you are in need of this medication it is just an odd thought which occurs as I attempted to refresh my own mind on this difficult subject.
Hi Don,
Is that the “badly written under-edited” and “short” (16 page) statement that credits nobody but Leo Smith for the transcendental ‘Gordian knot’ solution to the idealism/realism problem with which the author ‘wrestled’ in deep solitary meditation before finally finding enlightenment (along with Wikipeadia and Youtube)?
I ‘glanced’ at it, thinking the author was due the attention he gives to Galen Strawson, I was wrong. I believe the TPM reviewer was rather too charitable. I have also read all of the mini-essays the author has posted on this site. Along with the criticisms offered by Benjamin (who does well trying to hit moving targets).
As far as Kant and Schopenhauer are concerned I am indeed in need of the medicine found in good introductory texts. So thank you for the Warburton tip.
James;
Thank you for introducing this fascinating topic.
Don and Curious, I have enjoyed very much your discussion and Don please do not get discouraged by the number of comments. In my case, it was difficult to add something meaningful or different to your position. It is really hard for me to adhere to the Platonist/idealist position in this case. All my intuitions and formation point to numbers as a man-made thing. But as usual, I enjoy and learn a lot by analyzing the opposite position.
Would you all please help me understand it/ or point to literature in that regard? How do philosophers justify that number exists by themselves and are not man-made?
I read Mathematical existence by Kit Fine, but the paragraph criticizing the anti-Platonist position is troubling for me. I enclose the transcript:
“Nor does the alternative anti-Platonist position seem any more acceptable. If numbers are created, for example, there was presumably a time when they first came into existence. Perhaps it was when man first began to count. But don’t we want to say that there was once a time when the number of dinosaurs roaming the earth was over 1,000? And yet how can that be? There were plenty of dinosaurs alright, but not a single number; and so how could their number have then been over 1,000? The position also makes a mockery of how we wish to apply mathematics. Consider the following piece of reasoning: at some time in the past the number of male dinosaurs was over 500; at the same time, the number of female dinosaurs was also over 500; and so, at that time, the total number of dinosaurs was over 1,000. Here we appeal to the arithmetical fact that 500 + 500 = 1,000. But since were then no numbers, this simple piece of arithmetical reasoning cannot even get going.”
With all the respect for Kit Fine, this paragraph sounds odd to me. What am I missing? He is a Professor and I am not even a student.
The fact that there were no numbers in the Pleistocene because human did not exist does not mean that, after their invention, they cannot be applied backwards in a time when humans did not exist. This in my opinion is not proof of an ontological category for numbers. Again, what am I missing? I agree with Don numbers are a function of concepts. First, we need the concept dinosaur, and then we can apply a quantifier like a number. I believe that If human cognitive abilities did not result in pattern recognition and concepts, then numbers would not exist. Dinosaurs did not form concepts; therefore they were unaware of numbers. In my limited understanding of ontology, numbers would at best determine the quantity or proportions of what exists. It is hard for me to imagine that they exist in themselves. But I always like to be challenged I examine different positions.
Re:- jmiret | August 15, 2011, 12:39 pm
“Would you all please help me understand it/ or point to literature in that regard? How do philosophers justify that number exists by themselves and are not man-made? “
Possibly the following may help in this connection, Although I have not had time to read it in its entirety myself.
http://plato.stanford.edu/entries/platonism-mathematics/#EpiAcc
The only book I know of which includes a chapter on Numbers purely from a Philosophical outlook I Michael Jubian’s “Contemporary Metaphysics” available from Amazon or ABEbooks. To establish which Philosophers have written on numbers along the lines which currently interest us would be somewhat time consuming and difficult and even then their papers may not be readily obtainable unless one has some sort of connection with a University giving access to all books and Philosophical Journals. Possibly someone else here has a better idea than I.
The argument seems to hinge around whether you consider that the real world (TM) which apparently contains numbers, is actually a man made mapping of something else (which I consider is a good model) and therefore number arises in the mind of the beholder as a necessary part of dividing the world up into countable entities with size and number inherent in that process, or whether you consider that man and his Mind is IN the Real World (TM) and so the numbers are ‘just there’ ..since that’s where the mind is located anyway.
Hi JJmiret,
The argument by Fine that you point to is an old and apparently devastating one, at least when the target is Conceptualism or Psychologism – the claim that numbers exist (only) as mental objects. How it fares against other non-Platonist conceptions is another question.
I mentioned earlier the anti-nominalist John Burgess. He would respond to questions about numbers existing during the Pleistocene Era by saying that to talk in that way is to commit a kind of grammatical mistake. Like the Platonist he is committed to the existence of numbers but he is not committed to numbers being ‘timeless’ only to “the simple negative grammatical fact of the inapplicability of tense distinctions in mathematical contexts”. (If this claim can be made out then the epistemological objection to ‘platonism’ – the question of how we can know anything about objects that exist outside time and space – can perhaps be evaded.) The Internet Encyclopaedia of Philosophy also has an entry on ‘Mathematical Platonism’ (though the SEP as recommended by Don is generally better). And Platonism seems the starting point when it comes to looking for the truth about numbers whatever position you end up with…
http://www.iep.utm.edu/mathplat/
Curious, Don;
Thank you for the information, I am reading it, it is very instructive.
At the moment, I have 2 things in my mind. Are the conditions for number ontology existence, abstractedness and independence, completely different than for other ontologies? If so, why? Shouldn’t be there a general set of conditions for existence?
Second, I am still puzzled by the dinosaur story why is it so devastating? By the way, I interpret the story in this way. We, humans, say x dinosaurs existed but actually what truly existed was each individual animal. Our statement is useful for language and communication. I am still trying to figure out why the passage or argument was so devastating.
Hi Jmmiret,
It seems that what it is for the number 42 to exist (if it is an abstract object) is different from what the it is for the idea of “42” to exist and that in turn is different from what it is for a material object to exist.
If numbers do exist – and some claim they do not, that they are only fictions (nominalists) – it seems they might be mental objects (psychologism), collections of physical objects (immanent realism) or abstract objects (Platonism and other forms of realism).
In the Conceptualist scenario the number 42 exists – but only as a mental object that exists in the heads of some people (thus the idea of 42 is not distinct from 42). Some would say that to say this is just to say that numbers do not exist. But if you pursue that thought there are a lot of weird consequences (as Frege contended). Numbers that nobody has thought of do not exist, numbers spring into existence when you count high enough but disappear when nobody is thinking of them, if we become extinct “2+2=4” stops being true and before somebody first counted there was no number of dinosaurs, planets or whatever, All this seems to follow if you take it to be the case that numbers (and mathematical objects) are mental objects. The short Burgess paper I cited earlier is good on this.
http://www.richmond-philosophy.net/rjp/back_issues/rjp4_burgess.pdf
Conceptualism/Psychologism is, as I understand it, simply not a live option in the philosophy of mathematics and neither is immanent realism. The real argument is between those who say numbers exist as abstract objects (though some seem committed to the existence of numbers without wanting to commit to what they are exactly) and those who say numbers are fictions. Separate entries on mathematical fictionalism can be found here:
http://www.iep.utm.edu/mathfict/
http://plato.stanford.edu/entries/fictionalism-mathematics/
Re JJMIRET Aug 16th:-
Concerning your dinosaur problem in which you say there are problems in counting dinosaurs. It occurs to me that there are plenty of dead dinosaurs around currently and if a Palaeontologist unearths a group say five well preserved specimens there is surely no difficulty in counting them. What is the problem in inferring that so many millions of years ago there were five dinosaurs in exactly the same position. I grant you that there were no humans around in those times and the word and concept did not exist then, well it does not in my book as I am not a Platonist. So far as I know attempts are currently being made to establish the colour of dinosaurs I do not know how far this has progressed. So say we reach the conclusion that live dinosaurs were for the sake of argument, Green, do we have problem that the human experience Green did not exist at the time of dinosaurs I would say no. All we mean is had we been there, we would have seen five green Dinosaurs. There was nothing around at that time which had need of applying labels to groups indicating how many there were. I cannot accept that numbers were lurking somewhere awaiting do be discovered by Humans. It seems to me that if say a Dinosaur had five young to care for, and two went missing it would surely become aware the “number” had decreased it should have more than it currently had. So perhaps in a sense, Dinosaurs could count.
This problem of numbers reminds me of the response of some quantum physicists to questions by their colleagues concerning the incomprehensible aspects of the theory, for example Superposition. Forget it, just calculate.
You say “Are the conditions for number ontology existence, abstractedness and independence, completely different than for other ontologies? If so, why? Shouldn’t be there a general set of conditions for existence? “ My understanding of Ontology as it is currently used, is the efforts made by Humans to establish the nature of what might be the case outside of the human interpretation of Reality. In this connection I assume that there is only one ontology which all embracing. The overwhelming problem is that we just cannot get out of our heads to make observations. Assuming we could, possibly we would be confronted with something as alien and apparently incoherent as the above mentioned quantum phenomena.
Is there an argument I wonder, for saying Electromagnet radiation was occurring in Dinosaur days but there was nothing there to receive them, and numbers are accordingly the same?
Hi Jmmiret,
It seems that what it is for the number 42 to exist (if it is an abstract object) is different from what the it is for the idea of “42” to exist and that in turn is different from what it is for a material object to exist.
If numbers do exist – and some claim they do not, that they are only fictions (nominalists) – it seems they might be mental objects (psychologism), collections of physical objects (immanent realism) or abstract objects (Platonism and other forms of realism).
In the Conceptualist scenario the number 42 exists – but only as a mental object that exists in the heads of some people (thus the idea of 42 is not distinct from 42). Some would say that to say this is just to say that numbers do not exist. But if you pursue that thought there are a lot of weird consequences (as Frege contended). Numbers that nobody has thought of do not exist, numbers spring into existence when you count high enough but disappear when nobody is thinking of them, if we become extinct “2+2=4” stops being true and before somebody first counted there was no number of dinosaurs, planets or whatever, All this seems to follow if you take it to be the case that numbers (and mathematical objects) are mental objects. The short Burgess paper I cited earlier is good on this.
Conceptualism/Psychologism is, as I understand it, simply not a live option in the philosophy of mathematics and neither is immanent realism. The real argument is between those who say numbers exist as abstract objects (though some seem committed to the existence of numbers without wanting to commit to what they are exactly) and those who say numbers are fictions.
Don;
I completely agree with your arguments; I am not Platonists. I am just trying to understand the opposite position, and in the case of the dinosaurs example it appeared absurd to me. But then I read that people like Frege, Goedel and even russell among others believe in their existence. Whether i agree or not with them, my respect for them, for their contributions, compels me to make the effort. Even if after the process I keep my position, I would have learnt a lot and developed a balanced and more comprehensive opinion.
Regarding the Ontology problem, I also agree but in the case of numbers I am not certain how it will apply if their existence is not spacial. Perhaps from my biologists experience, first you define life and then you identify different forms of life. In this case, how can you do ontology without defining in some way existance? Is existance spacio-temporal? Is it independent of our mind? Is it abstract? etc I know thousands of years of eastern and western thought has been directed to these questions, but for me it appears hard to find something if you do not know what you are looking for.
Curious; thank you for your explanations. I am reading the references that Don and you sent. In fact I read them once but I realized that I will need to read them many times and follow references. It will take time until I grasp their position; it is difficult for me because I have never studied mathematics and logic at that level.
As with all these things, the way I look at this is simple.
Whatever existence is, is not something that is distinct and separate from us, as pontificators. In the limit.
The MOMENT we start to split the world into chunks and draw space-time maps to put the chunks in, is the moment we have utilised Number.
The argument then ensues as to whether Number is in the so split and divided realm, of materiality, or in our heads, or whether indeed our hads belong to that space time reality, or are stuck so far up our metaphysical backsides that we have ceased to employ language in a meaningful way at all.
Number is the way we relate to the world of phenomena. So far as we consider our minds as objective observers of that, number is in our heads, not in the world.
The confusion arises because we have come to regard that material world both as a distinct and separate entity from our consideration of it, yet deny that anything exists beyond it. Thus denying our own existence as objective observers!
This double think is ideal for Physics at a macro classical scale, where it is a very good approximation to many things, and gives lots of right answers. But it simply won’t do in an absolute sense, and gives rise to these silly puzzles.
The simple way out is to regard materiality as a reflection of something else, and us as semi-conscious beings, as the reflectors. The maths is in our heads, but is reflected in the material view we produce.
Whether you regard that view as profoundly spiritual, religious, platonic, supernatural, or just a different sort of science is of course down to you.
Numbers did exist for dinosaurs, because both concepts exist in our current world view. The past is the present – a story in a never ending present, at that.
Let’s say that the past and the future are theories to explain the present, and that theories we make up are full of phenomenal groupings of stuff, causality and number.
Gravity is a theory. We get that. Its as real as whatever it represents is real and its as real as its ability to accurately represent it, is real.
Go further. Causality and phenomena are also theories, time and space are theories. They are only as real as whatever they represent is real and they’re as real as their ability to accurately represent it, is real.
We live in a vast interlocking THEORY about the world, never in the real world. Mathematics is the distillation of the process of individuation. Which is something we have to have done in order to consider ourselves as individuals. Doing Mathematics is merely unravelling the mess we have got ourselves into, as is Science.
Leo,
It appears to me that there are several views in philosophy and science that address or have been addressing: the observer-in side observer representation-outside observer reality. In fact, in general terms, I adhere to this line of inquiry.
This is not new; has been proposed many times before, and is integral part of different schools of thinking.
Some of your comments sound odly familiar with some current neo-kantians, Lewis one of them. In fact they appear very similar. I have not studied him in depth but I migth try.
However, I have to say that I am inclined to move to Philosophy of Biology as a field of interest or hobby. It is closer to my formation, and more accesible to my limitations.
However, there are other people, very smart people that have outstanding credentials and a history of accomplishments, that have a different view.
I am trying to understand their view. Why they think the way they do, and I do not believe, at least for now, that the reason is so trivial. But it could be, I am just trying to find out.
“how can you do ontology without defining in some way existence? Is existence spatiotemporal? Is it independent of our mind? Is it abstract?.”
Hi JJMIRET
Earlier you asked: “Are the conditions for number ontology existence, abstractedness and independence, completely different than for other ontologies? If so, why? Shouldn’t there be a general set of conditions for existence?” I don’t know that I said much that helped you with this. I’m still trying to get clear on this myself, here are some thoughts mined largely from the SEP entry on ‘Abstract Objects’.
As David J. Chalmers puts it: “The basic question of ontology is ‘What exists?’ … the ontologist may ask: Do numbers exist? The Platonist says yes, and the nominalist says no.” Numbers appear in the ontology – inventory – of the (Mathematical) Platonist but do not appear in the list of things that exist to which the Nominalist is committed.
Abstractness is a quality that both the Platonist and Nominalist claim numbers possess if they exist. The claim that numbers are abstract objects is common and numbers are the paradigm case of abstract objects. Indeed the term ‘abstract’ was brought into its current philosophical use to deal with the mathematical objects that Frege proved could not be mental or concrete and which he contended must belong to some “third realm” (up to that point it had been generally thought these two categories exhausted the types of things that existed). The use of the term ‘abstract’ provides a contrast with the concrete. The abstract is largely defined by ‘Way Of Negation’ that is by referring to the properties possessed by paradigmatic concrete things that the abstract lacks. Most would say that it makes no sense to talk of abstract objects causing anything (this contrasts the abstract with what was claimed for Plato’s Forms). And it seems almost universally agreed that it makes no sense to talk of the spatiotemporal location of an abstract entity like a number. The strict physicalist seems committed to the claim that if x exists it exists at some spatiotemporal location, but whilst some physicalists are nominalists, most seem to make an exception for mathematical objects even if they aren’t entirely comfortable with it.
Independence, the third claim to which the ‘Platonist’ seems committed, is according to the IEP “independence of all rational activities, that is, the [mental or physical] activities of all rational beings.” I’m initially inclined to say that this amounts to the claim that there are numbers even if there is no one counting. Bringing in your question about mind independence and returning to the SEP entry on Abstract Objects “It is commonly supposed … that the game of chess is an abstract entity (Dummett 1973). But there is certainly a sense in which the game would not have existed were it not for the mental activity of human beings. So at least one sort of mind-dependence would appear to be compatible with abstractness”. Passing quickly by the talk of games …. Abstractness, it seems, does not necessitate Independence. And though the Platonist seems committed to both, it seems that an anti-nominalist like Burgess can assert that numbers “are abstract entities, to which it makes no sense to ascribe a position in space or date in time” whilst recognising that “mathematics is a human creation, since mathematics is a body of theory expressed in language, and language is a human creation”.
Returning to Chalmers and his paper ‘Ontological Anti-Realism’ but filling in some gaps, – “The basic question of ontology is ‘What exists?’ The basic question of metaontology is: are there objective answers to the basic question of ontology? … the ontologist may ask: Do numbers exist? The Platonist says yes, and the nominalist says no. The metaontologist may ask: is there an objective fact of the matter about whether numbers exist? The ontological realist says yes, and the ontological anti-realist says no.” Chalmers also notes ‘an intermediate sort of lightweight realism’ that holds that “while there are objective answers to ontological questions, these answers are somehow shallow or trivial, perhaps reflecting conceptual truths rather than the furniture of the world.” The IEP notes “anti-nominalists endorse ontological commitment to mathematical entities, but refuse to engage in speculation about the metaphysical nature of mathematical entities that goes beyond what can be supported by common sense and science”. It occurs to me that this sits fine with the idea that the claim that ‘numbers exist’ is really just a trivial conceptual truth.
Re Curious August 17th:-
I think we have to be careful here when we use the word EXIST. For a discussion to take place there has to be some agreement as to what what definition of exist is to be assumed. Chalmers has stated “The basic question of ontology is ‘What exists?’ … the ontologist may ask: Do numbers exist? The Platonist says yes, and the nominalist says no.” If the Nominalist says Numbers do not exist then we are apparently arguing about something which is not the case, and never has been, or will be. In fact there is nothing about which to argue. I am sure that nominalists do not support anything of that nature. It seems here that Chalmers may be suggesting that existence only embraces what we may call the physical world. This excludes things like Dreams, Ideas, Hopes, Joys, and Sorrows, Sherlock Holmes, and Mickey mouse. My suggestion is that existence is best described as anything that is in the nature of an event be physical or mental. The problems arise when certain mental events are denied existence because they have apparently no counterpart or extension in the physical world. For example dreams or fictional characters.
The problem with existence is is complicated. Replies to the question “What is existence”? Has generated a surprising number of answers both from philosophers of note, and others. It has been claimed that existence is property possessed by all individuals. Others say it it is a characteristic that some individuals have but others lack, for instance imaginary entities. Some have claimed that existence is not a predicate or not a characteristic or property of individuals at all. There are disputes concerning abstract objects like numbers and Universals, possible objects, and what are the fundamental constituents of Reality.
In view of the foregoing it does seem to me that an agreement as to definition should be sought before we make some sort of effort to consider Numbers and their existence.
Personally for me who considers that all I ever contemplate are my own ideas that is, a type of indirect realism, that we do not and can not perceive the external world as it really is but know only our ideas and interpretations of the way the world outside of our minds might be. This allows me to embrace possibly every thing as having existence in my mind, and in some cases additionally related to things which on the face of it seem to be outside of my mind but somehow seem to signal they are there by causing me to have thoughts about them although I can never examine them directly. This brings me to The Universe of Discourse for when I say Sherlock Holmes existed I mean that in the universe of Discourse which embraced Sherlock Holmes he did exist. In the Universe of discourse which embraces British prime ministers Holmes did not exist.
All I am trying to say here in somewhat too prolix a manner is that we must be clear on what we mean by existence and also what Universe of discourse we inhabit before starting an enquiry. For instance in the Universe of Discourse which is my mind Numbers exist. In the Universe of discourse of what may embrace what may be outside of my mind Numbers do not exist.
‘If the Nominalist says Numbers do not exist then we are apparently arguing about something which is not the case, and never has been, or will be. In fact there is nothing about which to argue. I am sure that nominalists do not support anything of that nature.’
Don,
That, apparently. is exactly what the nominalist claims – that numbers do not exist, never have and never will. As is mentioned, in the SEP entry you linked to, though it is possible for (mathematical) nominalists to hold a position called truth-value-realism – and thus agree that mathematical statements are true in the language of mathematics – they still maintain that “there are in fact no mathematical objects and thus in particular no numbers” in the language of philosophy. And it seems that nominalists have plenty to argue about – they have to argue against the Realists who hold the view that there are such things as abstract objects (as is noted in said entry “In contemporary philosophy nominalism is typically defined as the view that there are no abstract objects”).
The physicalist might indeed argue that ‘that existence only embraces what we may call the physical world’. For him there is no other realm of existing things, abstract or mental. Not all physicalists would assert that Dreams, Ideas, Hopes, Joys, and Sorrows do not exist (though of course there are those who claim such talk will be eliminated from a better perfected science). Many would simply qualify the claim that there are such things as ideas and dreams with the assertion that such mental phenomena are dependent on or reducible to physical objects. This is rather different from the case of Sherlock Holmes (as it were) and poor old Mickey Mouse. Pretty much everybody would agree that Sherlock Holmes, and Mickey mouse do not exist. And by this they do indeed mean that Sherlock and Mickey have no spatiotemporal extension in the ‘physical’ world – this is because Sherlock and Mickey are, as it were, purported concrete objects, for either to exist they would have to have spatiotemporal extension. But of course they do not because they are merely fictional concrete objects. There is a ‘world’ of discourse for Sherlock Holmes in which you can make truth-apt claims about his pipe smoking and so on but, of course, nobody is arguing over whether Holmes exists, ever existed or ever will exist.
Returning to the SEP, but to the entry on ‘Fictionalism in the Philosophy of Mathematics’ it is noted that “fictionalism is a version of mathematical nominalism, the view that there are no such things as mathematical objects”. Fictionalism, it is stated, “does not involve any very strong claims about the analogy between mathematics and fiction. For instance, there is no claim here that mathematical discourse is a kind of fictional discourse.” However whilst Fictionalism shares the general nominalist view “that (a) our mathematical sentences and theories do purport to be about abstract mathematical objects, as platonism suggests, but (b) there are no such things as abstract objects.” it seems to me to diverge from other forms of nominalism through its commitment to the claim that “(c) our mathematical theories are not true.” As is noted “When one first hears the fictionalist hypothesis, it can seem a bit crazy. Are we really supposed to believe that sentences like ‘3 is prime’ and ‘2 + 2 = 4’ are false? But the appeal of fictionalism starts to emerge when we realize what the alternatives are…”
After considering these alternative, the SEP entry conludes that “there are good reasons for rejecting the various anti-platonistic alternatives to fictionalism and, hence, for thinking that platonism and fictionalism are the two best views of mathematics, but there does not seem to be any good argument for favoring fictionalism over platonism or vice versa.”
“we must be clear on what we mean by existence and also what Universe of discourse we inhabit before starting an enquiry”
If I can return to Chalmers and his paper ‘Ontological Anti-Realism’. Therein it is metaontology that he is concerned to discuss and it seems it is metaontology that deals with these points you are entirely right to bring up. As mentioned Chalmers notes two main views in metaontology – Ontological Realism and Ontological Anti-Realism. Ontological Realism, he notes, “is often traced to Quine (1948), who held that we can determine what exists by seeing which entities are endorsed by our best scientific theory of the world.” This it seems is the majority position with regard to ontological claims – that existence claims about numbers are true or false outside particular ‘universes’ of discourse – that there is an objective fact of the matter when it comes to the question: “Do numbers exist?” . Thus if you say, as the fictionalist and other nominlaists do, that numbers do not exist you are a realist about numbers – just as much as the Platonist or the anti-nominalist is.
Ontological anti-realism, for which Chamlers argues, “is often traced to Carnap (1950), who held that there are many different ontological frameworks, holding that different sorts of entities exist, and that while some frameworks may be more useful than others for some purposes, there is no fact of the matter as to which framework is correct.” According to Carnap “questions about existence always involve linguistic frameworks: for example, the framework of mathematics, the framework of propositions, or the framework of commonsense objects”. There two sorts of existence questions – internal and external. One could ask the internal questions about the existence of, say, prime numbers within the discourse of mathematics and there will be truth-apt answers to such questions. However, for a philosopher to ask ‘Do numbers exist?’ as an ‘external’ existence question is really to pose a pseudoquestion to which there is no truth-apt answer.
The general drift of your comments makes me wonder if it ontological anti-realism appeals to you. But I don’t quite know what to make of your claim that “in the Universe of Discourse which is my mind Numbers exist” but that in “the Universe of discourse of what may embrace what may be outside of my mind Numbers” they don’t.
(Ps Apologies for the length of my posts in this thread – and the amount of quoting in them. I really am struggling to get to grips with something that is well beyond the breadth of my knowledge and the height of my abilities).
Re Curious 19th August.
Thanks for your reply. I was uncertain as to whether or not I should post my submission for 19th August I though it was not very well written nor clear, and tried to deal with too much. Having other things to do I thought just get rid of it and in a moment of desperation pressed the POST button; well that’s my excuse anyway.
You say Nominalists maintain that Numbers do not exist, never have and never will. My immediate response is to ask if that is the case then we have nothing to talk about. You cannot talk about what has never been. The problem here lies in my definition of existence. For me existence embraces all that has been in the world and all that is in the world. I include here human beings and their thoughts. Thoughts do not occur in the absence of interaction of organic material in the brain. I think the problem here lies in the fact it seems quite apparent to me that numbers are a human construct. Mental, but not physical, they have never been out there in the world in any physical sense or platonic sense. I cannot see how the most determined Nominalist can oppose this, but I may well have overstated the case here. In this connection One might ask the Nominalist If “Seven is less that nine” is true: how can that be so if seven and nine do not exist?
Why are numbers a human construct? Human being have an innate propensity for deduction and for that to proceed we need words which refer to things in the world, in the sense that some things are larger than others, some are smaller and some are equal. We need to say how much smaller or larger and so on? Numbers allow us to talk about Nature. You might also claim the logical operatives like “If—then” “either/or” “and” “if and only if” have no existence. Again these are expressions, or instruments or tools allowing us to talk about the world in the terms as to how humans perceive it. Much goes on in the human brain which does not emerge into the outside world but that is not to deny its existence. For instance Dreams are not spatio temporal in the usual sense of that expression, but they do exist. One could argue here that dreams do have a spatio-temporal aspect in that they are a product of certain neurological activity in the brain at a certain time. We could have a coherent conversation about Sherlock Holmes and as you say make truth apt comments about him just as if in the physical world he had spatial temporal qualities. I would claim in fact that I knew more about Sherlock Holmes than I do concerning American Presidents.
It seems my idea of existence is rightly or wrongly somewhat more embracing than that of many others. So for me, referring to so called Non-existent objects is not a failure to refer. If it is not a failure then for me any sensible or asensible object has existence be it not in what some call the real world. As you know there is so much literature concerning all this. Powerful arguments for and against. I have tried to avoid much background reading here making some effort to express my own feelings some of which I note does have the backing of higher authority and some of which does not.
Reading up on Fictionalism here I note that Fictionalism’ generally refers to a pragmatic, antirealist position in the debate over scientific realism. The use of a theory or concept can be reliable without the theory being true and without the entities mentioned actually existing. When truth (or existence) is lacking we are dealing with a fiction. Thus fictionalism is a corollary of instrumentalism, the view that what matters about a theory is its reliability in practice, adding to it the claim that science often employs useful fictions. I am sympathetic with this viewpoint but again the expression “actually existing” really needs definition for absolute clarity and that is where I find a problem.
I am in a quandary concerning your question as to whether Anti realism appeals to me. The short answer is yes, but when I look at the world from a scientific viewpoint it seems to me that Quine’s statement in his paper “On What there is” is very attractive, that is “we can determine what exists by seeing which entities are endorsed by our best scientific theory of the World.”
In matters of this nature I am always reminded Of Colin Mc Ginn’s paper The essence of which is to show we are currently, not sufficiently cognitively adapted or advanced to solve the Mind Body problem. I think this is equally applicable to the problems we are discussing here. Out of this my wavering between one or the other or maybe something else will continue. In this connection I now note your further comments of 19th.
Don,
Thank you for your comments. You raise a number of interesting points. However, it seems all else may as well be bracketed if you wish to insist that no human construction that has ever been thought of can be said not to exist. Would you really refuse to say that ‘God does not exist’ or ‘immaterial souls do not exist’ and only consent to the claim that the existence of such things may not be ‘in what some call the real world’?
Re Curious 22nd Aug:-
My idea rightly or wrongly, concerning existence does seem I agree, to embrace things which we would not usually say in the day to day usage of the word existence fall into that category. For instance in a day to day expression relating to existence we would tell a frightened child that ghosts do not exist. An adamant atheist might say that God certainly does not exist. This means that try as one may, a ghost or God will never be found in the daily conscious experience, as we examine the world with everything at our disposal, from our own senses, to scientific instruments as a means for discovery. I can understand this viewpoint and in the case of casual conversation may even speak along such lines.
However it seems to me that we can apply existence to things which it is abundantly clear do not exist in the world outside of our bodies. One obvious candidate here is Pain. It would be absurd for me to say I looked out of the window and saw some Pain there; Another candidate would be Love. In this connection we must not confuse the outward bodily demonstrations of Pain a and Love as actually being those to entities. Existence here is those innermost feelings which cannot be denied. Because something is not out there in the world so to speak is no reason to deny its existence. Possibly, to try to meet those who oppose this, I might use the word SUBSIST a form of existence as in a. to exist as a concept or relation rather than a fact or b. to be conceivable. Whether or not such entities have to be logical or not I am undecided. In this connection a belief can exist even if it be a false one.
It may be of interest here to say that certain entities are generally thought to exist in the outside world but do not. Examples here are Heat, Light, and Sound. All of these are Human experiences generated by forms of radiation emanating from objects.
Concerning the existence of God and or Souls I would say almost certainly these are not discoverable in the world as we currently understand it and as we understand the conventional definition of these entities. How ever we cannot deny that the ideas of of these do exist we can speak about them coherently even to the extent that we would if they did exist out there, wherever that is. The real world for me seems to embrace quite a lot. The outside world embraces concrete objects liquids solids and gaseous and all forms of projected radiation and would be there even in the absence of humans, so it seems. I am wondering now if all this merely devolves into a linguistic problem where all we need is an agreed definition of terms.
Don,
Thank you for your reply. There is a great deal worth thinking about in all of your last posts. I have had some trouble trying to organise my thoughts. Sometimes though I think we just have to press POST…
I don’t know that your claims for pains or love are central to your argument. I see no problem in contending that such things exist (and I share your background roughly physicalist realism). I rather feel that if you wish to employ the term ‘subsist’ it is better reserved for Pegasus, square circles and the like – let us not deprive our loves and pains of their place amongst the real! (As it were.) Thoughts, dreams, beliefs, notions, ideas and other non-spatial entities I welcome into our ontology – it is just the ‘things’ we find within them that I have reservations about marking down in our inventory. I am happy to concede that theists have some ‘idea’ or ‘notion’ of God – that is to say that they associate various inconsistent bundles of apparently false and incoherent beliefs to that name and others. But it does seem quite a step for me to accept that every bundle of foolish beliefs with a name attached to it is a ‘thing’ that ‘exists’ – except in so far as this is supposed to reflect the fact that very many bundles of false beliefs exist ‘inside’ the ‘minds’ of many people. But if I want to communicate that then it seems rather less misleading not to say ‘God exists’ but to say that ‘the notion of God persists’, or, better, ‘notions of God persist’. I do find theological debates oddly intriguing but I rather want to insist on my right to say that “God does not exist.”
I think your core intuition is that a thing has to exist if we are to make a true claim about it. We can state truths about “fictional” persons such as Sherlock Holmes – about his living at 221b Baker Street say – therefore Holmes must ‘exist’ (in some ‘Universe of Discourse’). You asked earlier how the Nominalist could hold that “seven is less that nine’ is true if he holds that “seven and nine do not exist.” The fictionalist, of course, says “seven is less than nine” is not true. But it seems those nominalists who are truth value realists can consistently accept an ‘internal’ ‘existence claim’ like “There are primes numbers between 10 and 20” whilst making the ‘external’ (to mathematics) claim ‘in the language of philosophy’ that ‘numbers do not exist.’ (Though, it seems, the ontological anti-realist takes this to be an invalid move he must presumably mark some distinction between Holmes and, say, Bertrand Russell.) It seems to me this can be explained in much the same way as we might talk of Sherlock Holmes outside of ‘The Game’. (This is the name of a ‘real’, and long ‘persisting’ phenomenon in which people take quite seriously the ‘gaps’ and inconsistencies in Dr Watson’s accounts of Holmes’ exploits and try to deduce the facts about where Holmes really was at given times in relation to calendars, reports of the weather and so on – quite an intriguing phenomenon I thought and seemingly, the source of far more pleasure than organised religion.) But in the universe (of discourse) that James and Don both inhabit it seems that Holmes cannot be found and, sadly, he is not in hiding. This is a pity as his deductive skills might have helped me here.
I gather ‘x does not exist’ remains self-defeating in your account and that you are in distinguished company when you say this and that others share your discomfort with talk of ‘non-existent objects’ (though presumably you would not contend that ‘non-existent objects’ do not exist). I am somewhat reluctant to condemn Holmes to ‘subsistence’ (it does sound rather undeserved) but it occurs to me that however we choose to use language we do not want to be committed to the claim that ‘Sherlock Holmes exists’ – unless we qualify that with “within the game”. It occurs to me that, in your account, to say “x exists” simply is to state “x is an object of thought.” And this it seems is made true by being meaningful – as long as “x” has some ‘content’ then “x exists” is true. And if “x” lacks any ‘sense’ it is just a cough followed by the word “exists.” To say “x does not exist” is self-defeating – it is rather like saying “this [object of thought] is not an object of thought.” To be is to be thought of, and to be thought of is to be. This doesn’t sit quite right with me but I don’t know that philosophical concerns about existence, subsistence versus non-existence objects and indeed numbers simply devolves into a linguistic problem where all we need is an agreed definition of terms. At times I think here I am simply stuck in the fly bottle. At others I think no, the puzzle does exist. And it is mystery worthy of great detectives such as Hume and Russell but I rather doubt Ihave the wit to follow their investigations.
*POST*
Re Curious:-
I continue to be dissatisfied with my own findings in this matter, You put your finger on perhaps my main worry when you said in you last post “However, it seems all else may as well be bracketed if you wish to insist that no human construction that has ever been thought of can be said not to exist. “. You are right and of course, there is something at fault here. The only way of getting round it so far as I can see is to accept that all I ever contemplate are my own ideas. Now what I mean here is that I do not have direct knowledge of the world. All I have is first hand knowledge of my own mentation much of which are events in my nervous system caused apparently by entities without my body. Thus for me existence is everything which is expressed by the firing of neurons in my head. This of course would include any old rubbish I might just fancy to dream up. Given any x in the world then x exists cannot be right. I don’t know who invented the word existence but I strongly suspect he/she did not mean what I have just said, which is in fact with small changes, a portal to the philosophy of Idealism. I must admit I am highly attracted to the supposition that we would have no access to a mind-independent reality even if it may exist What do we mean when we say something exists? The question itself presents some problems as we really need a definition by the interrogator as to what he means by something, because something can refer to I guess everything we could think of.
There seems no problem with statements Like “Elephants exist in Africa” or The Eiffel Tower exists in Paris. These are objects of being in the present. I agree with what you say about the subsistence of Unicorns and square circles. The reason why I would not condemn Holmes to subsistence is the fact that he could very well exist in what we might call the real world. For that matter I suppose all the characters in Dickens also could. The square circle obviously cannot exist in the world as we believe it to be, thus it subsides Perhaps I was not clear enough originally on subsistence.
You say “I do find theological debates oddly intriguing but I rather want to insist on my right to say that “God does not exist.” “ If what is intended here is that the concept of God the idea of him. her/it exists then I would agree that with those provisions God exists, However if what is intended is that God as a recognisable body of existence in itself, is or will be, somehow discoverable by some process unknown to me or unaccepted by me at present then my reply would be ‘That God almost certainly does not exist.’ I think I have mentioned before, that this is Richard Dawkins’ conclusion In The God Delusion. Like all good scientists he leaves the door just marginally ajar you never know what might happen. The regularities of nature upon which science depends are themselves open to question. So far we have done very well with this assumption but underneath the well known problem of Induction still lurks.
I note your doubt concerning the mental acumen to follow Hume and Russell, which I think is a problem shared my many, and many will not admit to it. I have a problem in following everything which is bedevilled by the fact that it take me ages and always has to read anything technical it is all so time consuming.
My final thought here is that there is something wrong with the word Existence
Don:
Good stuff!
The way I look at it is that nothing exists.
Or rather stuff exists, but only the human mind turns it into ‘things’
There is a hierarchy. Basic ‘things’ like grass, trees and so on are at the bottom.
Higher up we might have things like ‘wind’ ..does the wind exist? its a NAME for a group of similar experiences..
Higher up still we have things like ‘God’ which is a NAME for groups of similar experiences..maybe.
Maths is just one of the languages we have to separate and measure and discriminate. Do languages exist? In the ‘real world’
*shrug* its all a problem from logicians insistence that ANYTHING exists in the real world, beyond e.g. quantum soup, and that’s only another term of description…
The simple way out is that its ALL concepts that collect similar experience into categories. Mathematics is part of what does the collecting.And a concept in its own right.
Re Leo Smith.
Yes I don’t find much to a disagree with there. What bothers me is the underlying feeling that If I took the reins off my Philosophical thoughts I could quite easily devolve into a solipsist having no free will and that state of affairs I find attractive.
It was precisely that real and present fear that led me to my ‘third way’..
The stuff is not all my stuff and it isn’t mind stuff.
I have freedom, but its not complete. I have free will, but its not all powerful.
I simply put ‘Reality Stuff’ a step beyond the material world, whose (material world) rules are partly my interpretation of the stuff, and partly a transformation of rules that exist beyond my power to alter, in the Reality Stuff itself..
This works, for me, because I have free will, up to a point, and a much greater sphere of action than if the material world ‘was all there was’.
Where mathematics resides is not really interesting. It exists IN SOME SENSE.
I choose to place it in the transformational entity I call consciousness.
Where its effects can be noted in that entity’s production, the material world…
Don,
‘I find it of interest that this subject so far has attracted only five replies. Were it on Lesbians, Politics, or Ethics replies would flood in. Some hold that the only true philosophical subjects are Metaphysics and Epistemology.’
Just noticed 2,800 people thought the article was worth sharing though.
James,
Yes I have noticed this. I think I may have mentioned it previously. The most popular subjects are those which heretofore have interested me least. My inclinations are towards the philosophy of science and Metaphysics and Epistemology although I claim no great expertise in any. That said however I have persevered with trying to write something sensible in the other departments which has entailed some considerable research, and re reading long forgotten stuff by Mill and several others. I always did find Mill difficult to read and still it seems something of a task; its just his style. That said however I have built up a fair amount of knowledge in areas where it was lacking so that must be good. I will still keep trying but the charm of Philosophy for me, lies in the basic mysteries of how, and why, and could I really be a brain in a vat, or do objects perdure or endure?
possibly something to do with the difference between having a (received) _opinion_, and actually thinking…?