Mathematics and the Form of Life

‘It is fair to say that Wittgenstein brought about a revolution in philosophical perspective, making platonism – as traditionally presented – impossible. When a person can carry out a mathematical procedure correctly – for example, he can extend the series 2, 4, 6, 8…the platonist says he has learned to follow a rule or has apprehended a universal. We manifest our apprehension of a universal, our grasp of a rule, by at most a finite amount of behaviour, yet the grasp of a universal, or rule guarantees a potentially infinite amount of behaviour. Wittgenstein’s point is, roughly, that it is not because we have grasped a universal or rule that we behave in a certain way, but because we act in certain ways that we say that we have grasped a rule or universal. No rule or universal will guarantee that someone will not extend the series 2, 4, 6, 8, 17, 28, 1002, =, Δ…saying that at each step he is doing exactly what he did before. There is no such thing, Wittgenstein argued, as absolute sameness. A procedure is a case of doing the same thing again if it is a practice within what Wittgenstein called a ‘form of life’ – a community that shares perceptions of salience, routes of interest, feelings of naturalness – in which it is perceived as the same. If the platonist tries to step out of the form of life in order to tell those within how things really are, then he must come to grief. For outside the form of life there is nothing: no rules, no universals, no sameness, no reality.’ (Johnathan Lear, ‘Ethics, Mathematics and Relativism’, in ‘Essays on Moral Realism’, ed. Geoffrey Sayre-McCord, p. 87)

Thomistic-Wittgensteinian Concept Formation and a Problem For Naturalism

I’m reading Haldane’s and Smart’s debate, ‘Atheism and Theism’, and Haldane makes an interesting point regarding what he takes to be a problem for a materialist/naturalistic metaphysic – that of our formation of abstract and universal concepts – such as square or triangularity. He gives a quick look at the two more traditional options – innatism, where concepts are just, as the name suggests, innate. We’re just born with them.The other option is abstraction – where, again as the name suggests, we abstract our concepts from our experience with objects. He notes problems for each: on innatism, how many concepts are we born with, and why that many? How did they get there? Are we born with the concept of both square and rectangle? Why or why not? On abstraction-ism, he cites Geach’s argument:

‘In the late 1950s Peter Geach produced a powerful argument against this latter thesis [abstraction-ism]. The suggestion that the concept square, say, is acquired by experiencing a variety of square objects and attending to their squarenss, while bracketing their other aspects, is absurd because in order to attend selectively to the squareness of square objects you must already have the concept square: attending to an instance of a feature F as such, is to exercise the concept f.’ (p. 102)

Haldane proceeds to give an answer along Wittgensteinian lines – basically, our concepts are neither innate nor abstracted but taught. The later Wittgenstein more or else held that our language and understanding (and one may reasonably assume concepts to fall under those two headings) are developed, taught and shaped by our actually participating in life and non-linguistic activities – against, say Augustine, Wittgenstein holds that the public is prior to the private in language. As we become part of a community, we learn and acquire language from the community – so our concepts aren’t innate, since we have to be participating in the life of the community, and they aren’t abstracted, since by the same token the concept wasn’t available until it was taught.

Haldane then ties Aquinas in thus:

‘In order for something like the Wittgensteinian explanation to work it has to be the case that the child has a prior disposition or potentiality to form concepts under appropriate influences; it also has to be the case that there is one that is itself already possessed of the concept. Alice will not pick up the meaning of the term ‘cat’ unless she has a relevant potentiality, unless the structure of her receptivity is of the right sort. By the same token, that potentiality will not be actualized except by an intellect that is already active in using the concept, her older brother James, for example…here I am forging a link with Wittgenstein’s linguistic communitarian account of the origins of thinking in the individual, and that suggests diving these aspects of the intellect, at least in the first instance, between the teacher and the taught. In these terms one may say that Alice’s intellect is receptive to, or potentially informed by, the concept cat, while the mind or intellect of James who has already mastered the use of the term is active with, or actually informed by this concept. In teaching Alice the word, James imparts the concept and thereby actualizes her potentiality. This picture grants something both to innatism and abstractionsim. One the one hand, in order to explain possession of concepts a native power has to be postulated; but on the other it is allowed that, in a sense, concepts are acquired through experience.’ (p. 103)

The dilemma that Haldane sees for naturalism can be roughly stated as follows: given that neither innatism or abstractionism provide an adequate account of our grasp and use of concepts, something like the Wittgensteinian picture must be the case so as to avoid the horns of the dilemma. But if the Wittgensteinian picture is the case, then we have a problem of infinite regress: if the explanation of Alice’s conceptual ability is explained by James’ ability, then James’ conceptual ability calls for an explanation, and then that explanation calls for an explanation, and so on and so on. While the Wittgensteinian picture escapes the innatism/abstractionism dilemma, it opens itself up to the charge of infinite regress unless it can be shown coherently how concept-formation may have arisen.

Haldane forsees a possible way out by arguing for a kind of ‘fading conceptuality’ history of language:

‘…no history of thought or language can be philosophically adequate if it tries to meet the genesis problem by postulating ‘fading conceptuality’. Though it is not put in there terms, or indeed very often discussed at all, something of this sort is presumably part of a naturalistic versiuon of Wittgenstein’s linguistic theory. On this account the history of concept-formation and use is the history of language; a history that leads us back to pre-lingustic activities, back further to pre-mental life, to pre-replicating life and ultimately to pre-animate matter…what needs to be accounted for is a natural transition from the non-conceptual to the conceptual and that is not the same distinction as one between degrees of conceptual complexity. Doubtless Stone Age cave dwellers made fewer and less abstract discriminations than a contemporary physicist, but that is irrelevant; the point is that the ability to make any general classifications is a conceptual power.’ (p. 106)

While this line of argument isn’t a bullet-proof argument against a naturalistic theory of concept-formation, there do seem to be some genuine difficulties here.

Thoughts on Wittgenstein’s Philosophy of Religion

This is actually a comment I made here – but I’ve been wanting to do a post like this for some time, so I’m reproducing it here.

Wittgenstein’s philosophy of religion (if it can even be called that – his writings on religion are very scattered and don’t form one precise picture) is, more or less, fideism. This is likely a result of two things: first, his Roman Catholic education. John Haldane notes that:

‘First, Wittgenstein had been raised as a Catholic and in that period catechetics, the teaching of Catholic doctrine, favoured a question and answer style that derived from scholasticism but only gave abbreviated formulae and not arguments. He would have found this a betrayal of the religious quest and could not fail to have been reminded of it by the lists of questions and answers in the Summa. Second, In the first decades of the twentieth century there was a good deal of triumphalist Catholic apologetics in which people cited Aquinas as if he had an answer to everything and contained no errors or omissions. This again would have struck him as profoundly unphilosophical and also unspiritual.’ (from here, a brilliant interview)

Secondly, his love of Kierkegaard:

‘Kierkegaard was by far the most profound thinker of the last century.’ (As quoted in “Wittgenstein and Kierkegaard on the ethico-religious” by Roe Fremstedal in Ideas in History Vol. 1 (2006) {stolen from wikiquotes}

Taking the two of those together, it’s very easy to see why Wittgenstein takes the route that Christianity is more about practice than belief that can be rationally grounded in philosophical proofs (IE Aquinas).

I said on twitter that I have three main objections to his PoR – I’ll confine the rest of my comments to those three for brevity.

(a) A misunderstanding of Kierkegaard and (b) the relation of history to truth (Lessing)

Kierkegaard is profoundly misused by Wittgenstein – who builds on what he perceives to be Kierkegaard’s ‘leap of faith’ – i.e., the rejection of the need for rational proof the the affirmation of pure belief. Not only do we not need proof for faith, but faith as such isn’t even about ‘belief’ but about ethical practice.

The misunderstanding is this: Kierkegaard was not describing a fideistic ‘leap of faith’ but rather a rather stunning epsitemological move born out of wrestling with classical Christian ideas. I’ll let physicist/theologian T.F Torrance set Wittgenstein straight on Kierkegaard:

‘…(in ‘Philosophical Fragments’) Kierkegaard developed a sub-theme which turned out to have the greatest signifigance, the relation of truth to time, which had been conspicuously missing from Anselm’s thought. Behind Kierkegaard’s concern, as we can see from some of his other writings, lay his engagements with problems he found in Aristotle. Kant, Lessing and Hegel and the stimulation of some ideas he derived from Trendelenburg’s critique of Kantian notions of time. But what really gripped Kierkegaard and forced him to come to terms with it was the fact that in the Incarnation, “absolute” truth moved into time in Jesus Christ and became “historical fact”, which implies that we cannot know the truth except in a dynamic way involving a temporal or historical relation to it. If the truth has moved into time and become historical event, then movement or kinesis belongs to truth and has categorical significance.

In wrestling with this problem of transition Kierkegaard found he had to abandon a way of thinking from a point of absolute rest, and opt for a kinetic mode of reason with which to apprehend movement, continuty, dynamic truth, without resolving them into something quite different in terms of static necessities or timeless possibilites. He referred to his act of reason variously as a decision, a resolution or a leap, and spoke of faith as having the required condition.’ (T.F. Torrance, ‘Reality and Scientific Theology’, p. 90)

(c) Dictotomization (my new word) of practive/belief

Wittgenstein’s error here is an old one – assuming that things that are distinct are opposed. Practice is better than only belief, so Christianity has to be more about practice than belief. Had he simply payed close attention to the Christian tradition, he would have found that the concerns he had were more than addressed by Christian thought, though not in his language. The classical Christian ethical tradition holds that ‘doing the good’ requires ‘knowing the good’ because our actions have as a ‘formal cause’ our desires and beliefs. Desire/belief effects practice, and practice effects desire/belief. Virtue ethics is an appropriate reference point here.

To be sure, the Christian faith is about affirming certain truths – we could rephrase that to ‘making truth claims’ in modern lingo. However, as Torrance showed above, these aren’t static timeless propositions that one merely assents to – the knowledge effects the desires, which effect the actions (praxis), which in turn effects the desires. The ethical dimension of Christianity is far more than Wittgenstein’s ‘meaning is use’ move.

EDIT: I made a comment (scroll down if you’re on the post or click on it to see comments) here:  which hopefully clarifies and fleshes out some of what I layed out here.



On Meaning, Words, Games, and Problems

I recently read a conversation on Facebook about the conscience – this conversation went on for a good many posts, but it was ultimately an exercise in time-wasting. Why? Because terms were not defined. Not once in the lengthy thread were the terms under consideration defined or even really discussed. This means that the conversation was basically about nothing. It may as well have not have happened.

Sure, that’s a bit extreme of me to say, but I say it to illustrate what I believe to be the most important thing you can do in life: define your terms. What does X really mean? So much of what is said has no meaning simply because meaning is assumed. Don’t assume that the word has some innate meaning, because it doesn’t. But, the objection goes, then everything just becomes (as I typed the word ‘becomes’ I saw that I had typed ‘because’ instead and had to erase it) word games. Yes! Exactly!

It should be no secret to readers of this blog that Wittgenstein is my favourite philosopher, not because he was ‘right’ or whatever, but because of his method – letting the fly out of the bottle by kneading and working through the fogs and mists of our language to show us that the problems of philosophy really aren’t problems at all. I’m convinced that most problems, and not just in philosophy, are problems of language and meaning. By this I don’t mean I’m a logical atomist. I mean that our words and language games do more to hinder us than help us when we try and get to the root of a problem and that if we work through the game, we can often get to the real nub of the issue – maybe even to a solution. Maybe not, though – I don’t believe that philosophy is necessarily about trying to get to a set of certain doctrines. But if we can simply clear away or clear up the conceptual ground, perhaps we can discover that there isn’t really a problem after all – maybe we’ll even find an answer.

God and Logic

Logic, as Wittgenstein and Russell learned, is a very slippery thing. For starters, what is it? It’s not a physical thing – I don’t bump into logic on the street. It’s not necessarily reasonable – some things are perfectly logical while also being entirely unreasonable. But, oddly enough, like mathematics, we seem to be subject to it – regardless of its immaterial status. Is it odd that something immaterial is something to which everything material is subject?

One of the various ways in which John 1:1 can be translated is ‘In the beginning was the logic’. Logos has many shades of meaning – but suppose this is true, as Gordon Clark thought. Is logic uncreated, then? Is God logic? I say no – logic has far too many limits to be God. However, is God subject to logic? Can God do that which is illogical? If God can perform physical miracles, and is unlimited by the laws of nature, wouldn’t the same apply to logic? Can God make square circles? Married bachelors? I don’t think so – though part of me is tempted to say that if God decided to make a square circle, then we would simply have to redefine our notion of a circle. But generally, it’s agreed that God can’t do any of these things. Aquinas discusses this issue in a pretty definitive way in his Summa:

‘It remains, therefore, that God is called omnipotent because He can do all things that are possible absolutely; which is the second way of saying a thing is possible. For a thing is said to be possible or impossible absolutely, according to the relation in which the very terms stand to one another, possible if the predicate is not incompatible with the subject, as that Socrates sits; and absolutely impossible when the predicate is altogether incompatible with the subject, as, for instance, that a man is a donkey.’

‘Now nothing is opposed to the idea of being except non-being. Therefore, that which implies being and non-being at the same time is repugnant to the idea of an absolutely possible thing, within the scope of divine omnipotence. For such cannot come under the divine omnipotence, not because of any defect in the power of God, but because it has not the nature of a feasible or possible thing. Therefore, everything that does not imply a contradiction in terms, is numbered amongst those things, in respect of which God is called omnipotent: whereas whatever implies contradiction does not come within the scope of divine omnipotence, because it cannot have the aspect of possibility. Hence it is better to say that such things cannot be done, than that God cannot do them. Nor is this contrary to the word of the angel, saying: “No word shall be impossible with God.” For whatever implies a contradiction cannot be a word, because no intellect can possibly conceive such a thing.’ (I Q. 25, Art.3)

Interesting stuff. Aquinas was a sharp fellow.

What Can Be Said

4.116 Everything that can be thought at all can be thought clearly. Everything that can be said can be said clearly.

4.12 Propositions can represent the whole reality, but cannot represent what they must have in common with reality in order to be able to represent it – the logical form.

To be able to represent the logical form, we should have to be able to put ourselves with the propositions outside logic, that is outside the world.

4.121 Propositions cannot represent the logical form: this mirrors itself in the proposition.

That which mirrors itself in language, language cannot be said.

That which expresses itself in language, we cannot express by language.

The propositions show the logical form of reality.

They exhibit it. (Wittgenstein, ‘Tractatus’)

Here Wittgenstein makes some interesting observations: namely that proposition cannot picture the pictorial relationship between language and the world – the logical form is the pictorial relationship between language and the world. The standard example: a mother and daughter share a resemblance. You see the mother and see the daughter, but you don’t see a third thing called ‘resemblance’. That has to be shown – not said. To attempt to put into words the pictorial resemblance between mother and daughter (which must be shown) is to speak nonsense. It is something that cannot be said and must be shown.

This comes, obviously, from Wittgenstein’s early period – when he thought of language as strictly representing the world. Language obviously does much more than just represent the world – speech-acts, for example, don’t represent anything in the world – they don’t have a pictorial relationship with the world. Insofar as language does picture the world, as it obviously does, I think Wittgenstein is broadly right.

I detect some form of Kantianism here – we cannot gain direct access to reality through language/propositions because there is an aspect of reality which cannot be expressed by us in language but must be shown instead. This is the unsayable – ‘There is indeed the inexpressible. This shows itself; it is the mystical’, (6.522). The limit to the world of sense is the limit of language – we cannot go beyond language to reality. We seem to be trapped in language.

The obvious problem is that Wittgenstein’s whole project is nonsensical – fascinating, thought-provoking, brilliant, but nonsensical. He later came to realize this, by realizing what I noted above – that language cannot be restricted to only picturing reality. Language is much more subtle, complex and rich than that – it cannot be pigeonholed into such a narrow area without self-contradiction. If language was only used to picture reality, then Wittgenstein would be right:

4.001 The totality of propositions is the language.

4.01 The proposition is a picture of reality

‘…while it has been known for long before Wittgenstein that ‘we make ourselves pictures of fact” or that “the picture is a model of reality”, the real problem consists in the closer determination of the relationship predicated in the aphorisms, and it is at that point that the contentions of logical atomism turn out to be exceedingly restricting. This was recognized by Wittgenstein in his later years. The painstaking gropings of the Philosophical Investigations are a far cry from the self-assuredness of the Tractatus, where Wittgenstein claimed nothing less than to have formulated “unassailable and definitive” truths. As years went by, he came  to see that the full meaning of human discourse far transcends the realm of propositions that, as he put it, can be said clearly. Beyond what he called “surface grammar,” a “depth grammar’ emerged before his searching eyes. It was a discovery that made shambles of the sanguine hopes of his early days. “No wonder,” he remarked, ‘that we find it so difficult to know our way about.”

Wittgenstein on Certainty

194. ‘With the word “certain” we express complete conviction, the total absence of doubt, and thereby seek to convince other people. That is subjective certainty.

But when is something objectively certain? When a mistake is not possible. But what kind of possibility is that? Mustn’t mistakes be logically excluded?’ (Ludwig Wittgenstein, ‘On Certainty’)

I will confess right off the bat that I have no idea what Wittgenstein means by his final sentence here. Maybe I can come to some kind of understanding by working through this proposition.

What kind of fact logically excludes mistakes? The fact that I exist? On Certainty is a sustained mediation on (duh) certainty – the opening line is the famous Moore quote about knowing that there is one hand in front of you. Wittgenstein goes into a very long and in my opinion needless exploration of the mental workings behind certainty on topics such as this. This, I think, is misguided. My response is typically, what reason have I to doubt that my I exist, that I am real, that the world is real? That’s my typical response to what I would regard as scepticism beyond necessity – if one requires hundreds of pages of tortured groping for whether one can be certain that they, or the world or their hand exists, I suspect that there are more important problems for said sceptic to attend to.

However, to think a bit more on this notion: what is meant by logically excluding mistakes? On the surface, it would appear simple: something is certain when its negation is logically impossible. The world exists – this excludes the possibility of mistakes. We may be mistaken as to the nature of the world, or something along those lines, but we cannot be mistaken that there is a world, because if there wasn’t, there wouldn’t be. This seems to me to actually be quite unhelpful – Wittgenstein’s tortured method seems to be contagious.