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Month: September, 2012

The ‘Objectivity’ of Mathematics

The first thing I need to make clear is that I am not a mathematician.  I’m not so sure it is an area in which I could ever have a great deal of proficiency.  I have, however, become increasingly interested in the philosophy and psychology of mathematics.  Given my deficiencies in this area, though, I welcome comments even more than in any other section because, being in such alien territory, I feel I am even more likely to err than in other cases.

It was during my early reading that I had a debate with a friend:  I had argued that mathematics was not in fact objective and certainly no more objective than language, whatever that would mean.  The context of this was a discussion of how, I believe, we have launched strange items into space in an attempt to communicate or demonstrate that intelligible life exists on earth, and one such item consisted of a string of mathematical formulas.  I said this was misguided since it presupposed that another life would have mathematics, would have need of mathematics, which is entirely a human convention / cultural game.  My friend didn’t necessarily disagree on this point but did proffer that I had to agree it was, at the very least, more objective than anything language could offer as a means of communication.  And so our discussion began.

His point was, let’s say that you had three walnuts in front of you: no matter what word or sound uttered, the fact that, together, they comprise the concept ‘three’ is either very simple to explain or self-evident.

This part was relatively easy to dismantle.  You only need to look so far as South America (let alone extraterrestrially), to Daniel Everett’s work with the Piraha, to see that it is a cultural tradition, habit, convention, etc. to need to group items into classifications, that is, and that cultures can exist that have no need and therefore no use for such a convention in their own life patterns.  These people could not learn how to add cardinal numbers, despite asking to be taught, because they don’t use a calculus in their day to day lives:  even their trading system worked on concepts such as ‘fewer’ and ‘more’, ‘larger’ and ‘smaller’, without the need for breaking quantities down into units.
—  To retort that they are simpler, less civilised, less intelligent etc. would not be correct.  Their faculties are the same as ours but their way of living is so far removed from how other areas of the world have developed, their way of living so different, that what is relevant to them is vastly different from what is relevant to us, thereby shaping their entire way of thinking, speaking, living.  There is no reason to believe, or not to be able to conceive, that an advanced alien form of life could not function in a similar way–  or an entirely different one from either scenario, for that matter.

Consider Daniel Tammet, for instance, who has what he describes as synaesthetic experiences as opposed to anything we would call ‘calculating’.

The second part, however, caused me to falter.  He pointed to the rim of a circular drinking glass on the table and said:  “Look, a mathematical formula can calculate the exact surface area of this glass.  I can take the length from the centre to the edge of the rim, square it, multiply it by Pi, and I will discover the surface.  This formula will work on any circle.  It is axiomatic and will apply in any and every like case.”

I knew something wasn’t quite right with what he was saying, but I couldn’t pinpoint it.  I wanted to say:  You haven’t discovered anything.  The circumference or, more accurately, that rim shape was already there to begin with.  And this, I think, is the mistake being made.  Nothing has been discovered because the proof is already present–  the space defined by the rim, the length of the rim itself, were there from the start.  All that was necessary was to create a system that maintained consistency.  What more is needed?  If, from the very beginning, the formula for a circle’s area was:  Pi(r)squared – 1 , would this cause any problems for the engineers, architects, etc.?

Wittgenstein, Remarks on the Foundations of Mathematics, I.5., c.1937-1938:

“How should we get into conflict with truth, if our footrules were made of very soft rubber instead of wood and steel?– ‘Well, we shouldn’t know how to get the correct measurement of the table.’–  You mean: we should not get, or could not be sure of getting, that measurement which we get with our rigid rulers.  So if you had measured the table with the elastic rulers and said it measured  five feet by our usual way of measuring, you would be wrong;  but if you say that it measured five feet by your way of measuring, that is correct.– ‘But surely that isn’t measuring at all!’–  It is similar to our measuring and capable, in certain circumstances, of fulfilling ‘practical purposes’. (A shop-keeper might use it to treat different customers differently.)”

What is important to recognise here is the importance culture and custom play even in something as seemingly inexorable as mathematics.

“… it can’t be said of the series of natural numbers– any more than of our language– that it is true, but:  that it is usable and, above all, it is used.”  (I.4.)

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Morality and Why I Abandoned Moral Nihilism

I have been working on a piece concerning the philosophy of mathematics, but it’s not ready yet.  In the meantime, I thought I would choose a topic that started to interest me a couple of years ago that had formerly held no interest:  ethics.

For many years I felt comfortable with the premise that morality doesn’t really exist;  at least, not in an absolute or objective sense.  I suppose I subscribed to moral relativism to the extent that I didn’t think one culture’s code of ethics could be right or wrong for what could constitute the criterion of correctness?  All very well if you have a god to subscribe to but, for this atheist/naturalist/non-stamp-collector etc., that could never be an option.

Since studying philosophical grammar, ever from the Wittgensteinian perspective, I have had to abandon this, though; or at least to the extent of how I express my disposition towards morality.

I had a discussion with someone not so long ago who offered that atheism, for example, is a man-made concept and so does not really exist, whereas something like a leaf does–  it would exist whether mankind existed or not. 

All this really amounts to, though, is this:  A leaf is a physical object. Atheism is not.  One plays a different role, has a different relevance, in our lives to the other.

As I touched upon in my essay in my first post here, the fundamental mistake gripping the aforementioned person is the belief that a word corresponds to something.  This is not how language functions.

Take, for example, a proposition such as:  Does red really exist?

The problems arise when you try to say something philosophical;  try to express a deep meaning. By this, I mean, in doing this we are trying to examine our words in a context that causes confusion. E.g. when you were a child, or even now as an adult, if someone were to say to you ‘[your name], please could you pass me the red pen on your desk?’– you aren’t suddenly paralysed with confusion and doubt as to what exactly is being asked of you; of the philosophical/existential implications of the words used. You reach over and grab the red pen.

The mistake of thinking red corresponds to something: we didn’t learn of red through discovering ‘red’, as though it were a concept separate and distinct from any other object; it forms part of the grammar of how we distinguish between objects. Should you think you can simply summon the colour ‘red’ in your mind, isolated from any substance, is this red you’re seeing before your mind’s eye or is it amaranth? Or cerise? Or crimson? What could possibly count for a criterion of correctness here? A patch of paper or strip of paint? The use of this word is an intrinsic part of the grammar of our language and we use it meaningfully and effectively everyday in our lives until we force a scientific approach upon it and try to look for it, dissect it, find its essence etc.; apply a word such as ‘exist’ in a context that is completely unnecessary and illogical. It could make sense in a non-philosophical context: e.g.,

“Hey, we should use the colour Carmine– it would work really well in this room.”
–“There’s no such colour as Carmine!”
“Of course there is. Haven’t you heard of it?”
–“Absolute rubbish.”
“Carmine exists!”

It’s when you try to use the word ‘exist’ in some form of existentialist pseudo-philosophy that you fall into traps. Yes, yes, you can talk about the light spectrum and the effect on our eyes etc.; however, it doesn’t solve the philosophical issue because, suppose everyone in the course of time had always been born colour-blind, we would still have distinct rules for how to use colour-words and, let’s pretend ‘red’ as we know it wouldn’t exist to this version of humanity, does that necessarily mean it wouldn’t exist or that it just hasn’t been ‘discovered’!? But then, that would be madness because how would we have any form of reference to know whether our own current faculties are adequate or in fact impaired? And the entire spectrum is actually very different from how we see it?
— Irrelevant. Our words gain their meanings from their use, which is determined by our cultures, our way of life, our history of being, or what-have-you.

Bringing this back to ethics:  Our sense of right and wrong, of ethics, is purely psychological. But that doesn’t make the sensations you feel less real or, better phrasing yet, less relevant to your life within your culture than a pair of shoes you bought recently; hence why the word has meaning. Saying ‘ethics’ doesn’t really exist is nonsensical– we don’t say this in our everyday lives unless we are trying to exclaim something profound or philosophical. Ethical dilemmas present a different psychological state for everyone but ethics plays its role and demonstrates its presence through people’s horror to murder, abortion, theft, etc. within its culture.  Different cultures exhibit different reactions and so we comment on their different code of ethics;  or their lack of ethics, as some judge.

And that’s why I had to abandon my claim that ‘ethics don’t exist’, when expressed normatively.

***

Wittgenstein on ‘red’ (bold my emphasis):

PI58:

“I want to restrict the term ‘name’ to what cannot occur in the combination ‘X exists’.– And so one cannot say ‘Red exists’, because if there were no red, it could not be spoken of at all.” [Wittgenstein’s response to himself:]
— More correctly: If ‘X exists’ amounts to no more than ‘X’ has a meaning, then it is not a sentence which treats of X, but a sentence about our use of language, that is, about the use of the word ‘X’.

It looks to us as if we were saying something about the nature of red in saying that the words ‘Red exists’ do not make sense. Namely, that red exists ‘in and of itself’. The same idea– that this is a metaphysical statement about red– finds expression again when we say such a thing as that red is timeless, and perhaps more strongly in the word ‘indestructable’.

But what we really want is simply to take ‘Red exists’ as the statement: the word ‘red’ has a meaning. Or, perhaps more correctly, ‘Red does not exist’ as “‘Red’ has no meaning”. Only we do not want to say that that expression says this, but that this is what it would to be saying if it made sense– that the expression actually contradicts itself in the attempt to say that just because red exists ‘in and of itself’. Whereas the only contradiction lies in something like this: the sentence looks as if it were about the colour, while it is supposed to be saying something about the use of the word ‘red’.– In reality, however, we quite readily say that a particular colour exists, and that is as much to say that something exists that has that colour. And the first expression is no less accurate that the second; particularly where ‘what has the colour’ is not a physical object.

Debates on Religious Belief

Something I have wanted to write about for some time is the nature of debate, conversation etc. in regard to religious belief.  This was an area that struck me in particular when watching the debates between atheists and Christians/believers unfold on that creationist website;  there was an immeasurable gulf between the parties despite both believing that they were using logic as their common ground.  There are* numerous examples of debates on a matter that have ended in complete frustration and exasperation (more often for the atheists, it must be said) as a result of this inability to bridge the gap.

Unfortunately, despite being an intensely religious man, Wittgenstein devoted very little time to discussing religious belief, which I always found to be a shame given that I feel he could have shed great light on the issues at hand.  Having waded again and again through much of his writing in the last few years, I obtained Lectures and Conversations on Aesthetics, Psychology and Religious Belief, of which nothing contained within was in fact written by him.  It contains notes that several of his students had taken (much to his displeasure and annoyance) in seminars with him, conversations he had with friends etc., on the title subject matter.

Despite his lack of approval at these words now being read, they are very characteristic of him, consistent with his style, and, I feel, extremely insightful.  The main point I take from it, or, as Kripke has described in his own case, how it struck me, was that it’s often overlooked or, at least, underestimated how fundamental the differences are:

“Take two people, one of whom talks of his behaviour and of what happens to him in terms of retribution, the other one does not.  These people think entirely differently.  Yet, so far, you can’t say they believe different things.
Suppose someone is ill and he says: ‘This is a punishment,’ and I say: ‘If I’m ill, I don’t think of punishment at all.’  If you say:  ‘Do you believe the opposite?’– you can call it believing the opposite, but it is entirely different from what we would normally call believing the opposite.
I think differently, in a different way.  I say different things to myself.  I have different pictures.
It is this way:  if someone said: ‘Wittgenstein, you don’t take illness as punishment, so what do you believe?’– I’d say:  ‘I don’t have any thoughts of punishment.’
There are, for instance, these entirely different ways of thinking first of all–  which needn’t be expressed by one person saying one thing, another person another thing.’
What we call believing in a Judgement Day or not believing in a Judgement Day–  The expression of belief may play an absolutely minor role.
If you ask me whether or not I believe in a Judgement Day, in the sense in which religious people have belief in it, I wouldn’t say: ‘No.  I don’t believe there will be such a thing.’  It would seem to me utterly crazy to say this.
And then I can give the explanation: ‘I don’t believe in…’, but then the religious person never believes what I describe.
I can’t say.  I can’t contradict that person.
In one sense, I understand all he says–  the English words ‘God’, ‘separate’, etc. I understand.  I could say: ‘I don’t believe in this,’ and this would be true, meaning I haven’t got these thoughts or anything that hangs together with them.  But not that I could contradict the thing.
[…]
“You may say they reason wrongly. 
In certain cases you would say they reason wrongly, meaning they contradict us.  In other cases you would say they don’t reason at all, or ‘It is an entirely different kind of reasoning.”  The first, you would say in the case in which they reason in a similar way to us, and make something corresponding to our blunders.
Whether a thing is a blunder or not–  it is a blunder in a particular system.  Just as something is a blunder in a particular game and not in another.
… If they do something very like one of our blunders, I would say, I don’t know.  It depends on further surroundings of it.
[…]
“Suppose someone dreamt of the Last Judgement, and said he now knew what it would be like.  Suppose someone said: ‘This is poor evidence,’  I would say:  ‘If you want to compare it with the evidence for it’s raining to-morrow it is no evidence at all.’  He may make it sound as if by stretching the point you may call it evidence.  But it may be more than ridiculous as evidence.  But now, would I be prepared to say:  ‘You are basing your belief on extremely slender evidence, to put it mildly.’  Why should I regard this dream as evidence– measuring its validity as though I were measuring the the validity of the evidence for meteorological events?
If you compare it with anything in Science which we call evidence, you can’t credit that anyone could soberly argue: ‘Well, I had this dream… therefore… Last Judgement’.  You might say: ‘For a blunder, that’s too big.’  If you suddenly wrote down numbers on the blackboard, and then said: ‘Now , I’m going to add,’ and then said: ‘2 and 21 is 13,’ etc. I’d say: ‘This is no blunder.’
There are cases where I’d say he’s mad, or he’s making fun.  Then there might be cases where I have to look for an entirely different interpretation altogether.  In order to see what the explanation is I should have to see the sum, to see in what way it is done, what he makes follow from it, what are the different circumstances under which he does it, etc.”

If you have read the above, I sympathise in you perhaps thinking he sounds mad, or talks gibberish, etc. but on this I would stress again that, whilst characteristic of his writing style, refrain from too critical a judgement since the context was him speaking aloud, following a train of thought, rather than trying to construct a specific, coherent argument.  The parts I have highlighted in bold are what I consider key.

The mathematics example is perhaps the simplest to follow:  The point is that there are well established rules and a common ground for logical inference as in this case.  In a mathematical equation, it is often very easy to see where someone has erred in a calculation:

“You forgot to carry the three at this point, which is why at the multiplication stage…”
— “Ah!  You’re right.  My mistake.”

If someone were to add 2 and 21 and have the result of 13, would we know where to begin looking as to where they have gone wrong?  They both believe they’re playing the same game.  This person might even have their own consistent rules and function perfectly happy with their grasp of mathematics for their purpose.  To get to the root of the difference, it is necessary to go right back to the beginning, to examine the very first foundations they formed for having this view;  this view that is worlds apart with your own.

My emphasis on this is that, when getting caught up in (a)theistic debates, often both parties try to justify their position, or corrode the other’s, by appealing to mutually acknowledged logical principles.  For some, however, the differences are so fundamental that it affects their entire outlook, their entire way of viewing the world.  What’s needed is to examine the bedrock from which their beliefs have developed.

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

*Unfortunately, the ‘Question Evolution Blog’ editors are in the habit of deleting any arguments presented that have any strength at all and so a lot of what might have proved interesting to anyone wandering through the site is lost.

Pascal’s Wager

Having recently heard of Ma Nuit Chez Maud,  in preparation for watching it, I reacquainted myself with Pascal and this of course led to re-reading the infamous wager:

1. “God is, or He is not”
2. A Game is being played… where heads or tails will turn up.
3. According to reason, you can defend either of the propositions.
4. You must wager. (It’s not optional.)
5. Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing.
6. Wager, then, without hesitation that He is. (…) There is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite. And so our proposition is of infinite force, when there is the finite to stake in a game where there are equal risks of gain and of loss, and the infinite to gain.

I thought this merited a quick blog entry because it epitomises the very problem I keep reiterating:  the one of bivalency.  My main contention, of course, is with the first two premises.  The very first proposition means nothing to me–  I want at this point to say:  “Why?”… “How so?”… “What exactly do you mean by that?” … “How did this “god” you speak of enter the equation?”

Notice the huge leap in the second proposition, where it is analogous to flicking a coin–  a misleading comparison.  After all, for either the tails or heads outcome, it is dependent upon both already existing and an established rule/acceptance that each outcome has an equal possibility. 

1. “Glorb is, or It is not.”

Or: it’s sheer nonsense.

Not everything can be assigned a truth value.

Is Atheism a Religion?

This is a question that is often posed.  I found a post from ‘Atheist Dave’ on the matter and saw some very interesting responses in the comments section.  The main source of interest was the division it caused not only amongst the religious and the non-religious, but even between the latter themselves.  I find myself up and typing in the early hours of the morning at the moment and so I’m simply going to type my rambling thoughts as they strike me, throwing structure to the wind.

I appreciate my views are slightly off pieste regarding typical atheist reasoning;  however, I thought I would chip in my two cents on the matter, even at risk of sounding like a broken record, but I always worry I’m not expressing myself clearly.

In my concept of atheism, I don’t see how it possibly could be seen as a religion.  I mean this in the sense that I imagine the person who is born and grows up having no exposure to religion, to the concept of ‘God’, and how they live their life and view the world.  Conceivable?  It is difficult to imagine.  Hypothetically, though, I can see this person marvelling at things such as space, causation, beauty, etc., and not attributing them to what we call ‘god’;  and by that I mean the concept of ‘god’ that imagines an omnipotent, omniscient– or perhaps only cognitive– supernatural entity.  This is how I would have the atheist view things;  without the social, cultural, and linguistic baggage that accompanies the word/concept.

I want, as stated in my essay, to imagine someone approaching my hypothetical person (although, in reality, this means: approaching me) and to say:  ‘So, do you believe in god?’ or ‘Are you an atheist or a theist?’;  and for the puzzled response to be:  ‘Excuse me?  What are you talking about?’;  ‘I’m sorry, I don’t quite follow you.’

They then proceed to say:  ‘You know… GOD.  Our creator, the creator of this world, the being responsible for […]’ etc. etc..  I would imagine or hope the response to be:  ‘What wild speculation!  That’s a very imaginative idea, though.  Whatever led you to make such a bizarre supposition?  And what led you to choose one over many?’ etc..

What could lead us to see this ‘god’ concept as credible as opposed to just one type of speculation amongst countless others?  Why couldn’t ‘glorb’ be an equally viable alternative?

The argument I noticed most commonly being reiterated by those claiming that atheism certainly is a religion went along these lines:

“You are denying the existence of God;  you believe he doesn’t exist or, as some say, know he doesn’t exist and therefore, without evidence, yours is a position of great faith.”

Hopefully, by now, my response will be predictable (and I apologise for being a broken record):  I can’t deny something that I have no idea about.  What is this ‘god’ you speak of?  You might as well say that I’m denying the existence of ‘glorb’, ‘molarb’, ‘trialanksipoke’, etc..  Whatever you’re speaking about doesn’t feature in my world.  You say, for instance, that everything has to have a cause and that ‘god’ is the origin of the chain of causation, but why couldn’t that cause have been a small bubble popping, from which everything followed?  Or, given we’re talking about a state of physics about which no-one has anything to go on aside from speculation, why does a chain of causation need to be finite?  That there needs to be a first cause?

Hume on causation:  a human belief;  one that is based on repetition;  inductive knowledge.  A rubber ball bounces where a lead one doesn’t because, having dropped both once, to check that the results are consistent, we drop them both several times further to see whether a rule can be formed.  Did the rubber one bounce as a freakish one-off event?  Or should the lead one have bounced but, for some reason, on this occasion, it did not?  All of our rules about physical properties are governed in this way.  How could we possibly apply them to something so grand and, as yet, unverifiable as the origin of the Universe?  There is no reason it could not be an exception to the rule. 
“It would be inconsistent with what we know.”
—  What we ‘know’ is based on our every day experiments with / uses of the properties to be found in and around our planet;  nothing else.

A.C. Grayling’s example, that I noticed Atheist Dave borrowed in his opening post on the subject matter, is right on the point:  If atheism is a religion, then ‘not-collecting-stamps’ is a hobby.

My own spin on this, I suppose, would have to be:  If atheism is a religion, then not-collecting-glibabs must be a hobby.  Or something.