The Philosophy of Logic and Mathematics

Hello.

So I've had the pleasure of having both of my HCJ seminar papers this academic year being on the topic of Logic and the Philosophy of Mathematics and in particular the works of Gottlob Frege and Bertrand Russell.

I might have described myself as agnostic before this injustice was visited upon me, but now I'm a solid theist who believes in a God that takes a perverse interest in dull human affairs and meddles accordingly. A God that just wanted to see me stutter my way through grotesquely complex material that I'll never, ever understand.

I'll try to write about it anyway in spite of the fact that I don't understand it. Wittgenstein's “whereof one cannot speak, thereof one must be silent” can stay on the shelf for now.

Logic and Technology

Logic is a means of working out a statement's truth value. Something can be true, false, or meaningless (if there is no means of verification.) Human beings are veritable verification machines – we approach everything in a logical manner, whether it looks that way or not. We're so complex that we don't even have to really think about the processes that we go through to reach our conclusions – it's only in the last hundred years or so where men like Frege have attempted to map this out, that technology has been able to advance forward and mimic these processes.

'Intuitive' has become a buzzword for lauding consumer electronics or computer programs on their usefulness recently, and when you think about it, that's an absolutely ridiculous premise to judge a computer on when the concept of 'intuition' is a very human thing and is, of course, constrained by the people who programmed it in the first place. You might as well judge it on whether you think it has a good sense of humour or not. It is only as computers reach ever increasing levels of complexity that even begin to approach the human brain that these gadgets have become 'intuitive' and therefore pleasant to use instead of incredibly infuriating – I remember my dad had a Palm Pilot (this is seriously going back) when I was a tiny little girl and it never did what you wanted it to do, purely it was limited by its own rubbish hardware and programming. This doesn't really happen so much any more. And that's great, and a testament to how far everything has come along.

When talking about how technology has progressed, that's just shorthand for how far the ability for us to express and map out inconceivably complex logic has come along.

Frege

Gottlob Frege was a German philosopher and mathematician. He was among the first to truly define numbers as 'the expression of pure deductive logic' and remove the pervasive idea of numbers being seen as mystical entities since the time of Pythagoras. The number 3 held particular significance to Pythagoras and his followers in the form of 'the golden ratio', something that was applied in art and architecture.

The de-mystification of mathematics helped to pave the way for the idea that all philosophical reasoning can simply be reduced to its series of core arguments and analysis of the way in which they have been put togther (syntax), and this is the underpinning for analytical philosophy. Frege showed that all arithmetic was could be broken down to a set of simple, logic premises. He also introduced new logical notation that could express generality, such as 'all' 'some' 'none' and 'if'. Frege maintained that the same rules that apply to numbers can apply to language, as long as the language used is clear and precise.

Bertrand Russell expanded upon what Frege had already built. Frege's work mainly concerned the logic of language, whereas Russell was trying to prove that mathematics should hold a inexorable link in logic as, much like music, it is a 'perfect' language. There are no connotations, no subjectivity – a number 9 is a 9 and a D sharp is a D sharp (or an E flat).

Something humanity had always struggled with was trying to pin down exactly what numbers are; if they're not mystical entities or Platonic perfect forms, then it stands to reason that they are empirical generalisations of what you usually get when you add objects together: 1 bead + 1 bead + 2 beads, which is fine, but that breaks down when you add 1 drop of water to 1 drop of water as you are still left with 1 drop of water.

Russell instead took the lead from Kant and put forward the notion that numbers are synthetic, a priori propositions that can be defined logically from a simple, limited set of axioms. Axioms according to Russell are self-evident, logical truths.

Principia Mathematica

Russell's Principia Mathematica was an attempt to show that mathematics is simply a follow on from logic. He believed that the philosophy of logic enjoyed a high status and that it was only fair – given that mathematics was made up from component pieces of pure logic - that maths should have an equal status.

He began by defining the vocabulary for logical premises i.e. X and ¬X (the sky is blue and the sky is not blue) to stand for propositions, and then gave examples of how you would use that vocubulary in practice. The syntax used to make 'deductions' is expressed as the following:

If X, then Y.
X.
Therefore Y.

So that could be something like:

If it rains, then the ground becomes wet.

It is raining,

Therefore the ground is wet.

He then moved on to the problems surrounding numbers, as mentioned earlier. The main problem is how to define a number without referring to the concept of a number itself – the same issue that would arise if you were to try and explain what 'red' is to someone who can't see. It can quickly become very circular.
The solution was to think of numbers in terms of 'sets.' The number 2 is simply a way of saying 'the set of all sets of couples', in the same way that number 3 is 'the set of all sets of trios.'

Notation has since developed to the point where the Principia is considered to be unreadable to 'amateur' logicians and people have pointed out inconsistencies in the work. Russell, for the sake of his quest to derive complex arithemetic from 'self evident truths' (axioms) was forced to introduce two axioms to the reader - the axioms of infinity and reducibility. Critics noted that neither were a priori, and required experience to measure and prove.

Additional notes from lecture:

Natural numbers are special words used in counting – the act of counting is to create a group of items.

Apes and stone-age tribes have types of very simple number systems that relied on natural numbers, and the extent of counting would consist of: “one thing, “more than one thing” and “many things.” They didn't need to go in to any further complexity and so these were the only numbers they needed.

Even for people from advanced cultures, small number words are functionally different from large number words. Most people can instinctively know that there are up to seven objects in a group before having to step aside and physically count them; you could say with conviction that there were only three people in a room, but you could not say that there were 37,498 people at a football match without some means of proving that.

Addition and multiplication are plurals of plurals. Creating words and abstract symbols for plural categories requires a system of number words (again symbols) and a logical syntax for combining those number words to imply further or predicate number words. In this instance, the number 7,367 is a predicate symbol of a bunch of more basic symbols organised according to a known syntax; the point is, you can't conceptualise what 7,367 looks like without breaking it down in this way.

A predicate, in logic, can be analysed (similar to division and calculating number squares of roots in mathematics.) This statement is the basis of analytic philosophy, on which all modern technology bases itself. Computers follow a basic logical language – an ongoing debate is whether computers can ever reach the complexities of human logic.

Ancient civilisations used hieroglyphics for number words – Greek and Roman societies depended on numbered symbols and their system did not regard zero and one as numbers.

The introduction of the concept of 'zero' came from India, and was absolutely ground-breaking – the idea that you can describe 'nothing' had caused philosophers considerable agony for millennia. This was because the proposition that zero = nothing in itself meant that it was something. It went against the Law of the Excluded middle formulated by Aristotle – this is the idea that for any proposition, it must be true, or that its opposite must be. It excludes any other possibility and zero, not being either anything or nothing, did not fit with this rule.

 

Seminar Paper on Logic

I didn't realise that I was supposed to focus on Principia Mathematica but here it is anyway.
 

Seminar Paper – Mathematics, Logic and Language

History of Western Philosophy, Chapter 31:

Russell wrote a body of ground-breaking work in the field of logic up until the age of about 40, after which he took a bit of a breather and wrote light-hearted stuff like the History of Western Philosophy instead. In his chapter on The Philosophy of Logical Analysis, he speaks about the intellectual divide between men who are more influenced by empirical sciences and men who are mainly inspired by mathematics.

He describes Kant and Plato as being among the empiricists; Locke and Aristotle belong to the mathematician party. Russell says that modern philosophy sets out to remove Pythagoreanism from the principles of maths, and to combine empiricism with the deductive parts of human knowledge.

Georg Cantor worked on 'infinite' numbers – and says to think of them in terms of a collection, and that an infinite collection is one that has parts containing as many terms as the whole collection contains. The Mandelbrot set is a visual representation of this idea; infinitely recurring self-similar fractal/

Russell says that it was Frege that truly defined numbers. He said that in the past, they had been clumsily assigned the meaning of their own plurality – as seen in the Pythagorean idea of the number 3 containing a triad, some kind of mystical property. Frege went on to demonstrate that mathematics is nothing but a prolongation of pure deductive logic. This helped to pave the way that most philosophical reasoning can be reduced to syntax. Rudolf Carnap thought that all philosophical problems were really syntactical, and that by trying to avoid problems with the syntax you either solved your quandary or rendered it insoluble; an extreme view perhaps,

Russell claims that the problem regarding accidentally assigning existence to an impossible object i.e. the golden mountain can be properly circumnavigated simply by using good syntax. “There is no entity c such that 'x is golden and mountainous' is true when x is c, but not otherwise”

He goes on to say that, though maths is not empirical, it is neither strictly a priori knowledge about the world; he argues that it is merely verbal knowledge and does not deserve the 'lofty' place it has been given. Offers physics as the best level of understanding one can have about the world – philosophers can now think of particles occupying a space in space time, and speak of quantum 'events' instead of just particles.

He also argues that the deeper understanding of physics and the physiology of the way our minds perceive events throws 'a new light on the problem of perception.' He tries to definite the word 'perception' – it must be an effect of the object perceived, and must more or less resemble the object if it is to be a source of any knowledge of said object. He says the the first condition can only be fulfilled if this effect is adhering to laws which are independent from the rest of the world – that is, in terms that I can understand, an effect which is scientifically demonstrable in a controlled environment. Physics can back this up – 'seeing the sun' for example, can be verified by study in to light waves and the biology of the eye, so that we can prove that we are 'seeing' the sun. Russell goes on to say that despite this, our personal knowledge of the physical world is merely abstract and mathematical.

In conclusion, modern analytical empiricism is for Russell a way of separating the search for truth and the search for ethics and value. The muddling of the two branches of philosophy have significantly impeded human progress as people have attempted to impose their own beliefs in philosophy and bent the rules of mathematics and logic in order to validate themselves; St Thomas Aquinas, Kant, etc. He believes that logical analysis is the best way to uncover truth, and that truth is the only thing a philosopher should concern themselves with.

Philosophy in the Modern World:

John Stuart Mill was the first British empiricist to work on formal logic. He rejected Hobbes' version of nominalism, which is the two-name theory of proposition – the theory that a proposition is only really true if the subject and predicate are the same thing, e.g. Superman is Clark Kent. Mill said that this only worked where the subject and predicate were both proper names. He developed the idea that a name could be pretty much anything – any word that contains the attributes of the subject it is trying to describe – therefore, 'wise' and 'old man' could denote 'Socrates' as part of a proposition. He accepts that every proposition is, in effect, a conjunction of names – though 'name' in its broadest sense as explained before. Although this sounds pointless, it is the connotations in these names that go about trying to establish a truth value – the proposition 'all men are mortal' tells us that the attributes of animality and rationality found in men are always accompanied by the attribute of mortality.

Mill wrestles with the idea that the conclusions of many propositions are actually contained within the premiss, and that no new knowledge is actually being derived from them. He thinks that syllogistic logic does not contain a genuine inference.

Induction is the process of moving from the specific to the general – deriving a certain truth from a set of particular instances. Mill says that inferences can be deductively valid without necessarily being informative or bringing to light any new knowledge, which is the nature of most syllogisms. He is interested to know how human beings can make the connection between two connoted attributes (like in 'All men are mortal'.) (A machine would think that the class 'men' also belong to the class 'mortal')

He set out five 'canons' to guide the inductive discovery of cause and effect; the first two are:

Method of agreement – If phenomenon A occurs in the presence of A, B and C and also in C, D, and E, then we are to conclude that C – the only common feature – is causally related to F.

Method of disagreement – if F occurs in the presence of A, B and C, but not in the presence of A, B and D, then we have to conclude that C is causally related to F as it was the only feature differing in the two cases.

Mill thinks that we are always applying these canons to everyday life: If we hear that a man has died after being shot in the heart, then we conclude that it was the gunshot that killed him because he was in perfect health before the wound was inflicted upon him. The wound is the only circumstance that has changed.

Frege attempted to show that all arithmetic was deducible to a purely logical premisses, and his logical methods were the polar opposite to Mill's. Mills conclusions were drawn from a posteriori propositions, whereas logic for Frege was a priori and also analytic.

Frege developed a system to overcome the weaknesses inherent in syllogistic logic that had been uncovered over the past couple of centuries – problems such as the syllogism only being designed to cope with subject-predicate sentences, and being unable to account for inferences in which words like 'all' or 'some' were included.

He replaced the ideas of subject and predicate with the words argument and function instead, which allowed grammarians and logicians a more flexible way of analysing a sentence. The boundary between what is the argument and what is the function can be moved in the sentence, so 'Wellington defeated Napoleon' can be analysed in a number of different ways. It gives them room to mark out logically relevant similarities between two sentences.

Frege also added new notations to logical sentences in order to express generality. These are symbols which are added to the structure of the proposition in order to mark out that a certain function is always true or false no matter what the argument is; this include notations for 'all' 'none' 'some' and 'if', which was arguably the most important element.

To make a statement like this, logicians would write: (x)(x is mortal) ...which is technically false, as it means that everything, everywhere is mortal, and Frege believed that all objects were nameable; therefore, this statement is attempting to say that the number 10 is mortal, which is wrong.

Existentialism

Existentialism – Take Two

We've briefly looked at existentialism before last semester, but not in the capacity of the effect it has on politics.

This time, it wasn't about some deep philosophical debate on how one should feel if, in the words of those amazingly profound lyricists from Linkin Park, “nothing really matters.”

We instead focused on the ramifications of nihilism and existentialism when those thoughts occur to the average, downtrodden guy – if you're being oppressed, rise up and crush your oppressor. There's nothing to lose. If there truly is no universal moral code, and no eternal metaphysical consequence for your actions, who's to stop you from beating your slavemasters and their families to death? You have nothing to lose but your own dignity if you continue to allow yourself to be a victim. If your existence can only ever be as meaningful or meaningless as you choose to make it, then the only way you can let yourself down is to not make that choice in the first place. You have nothing else to answer to.

If you're reading this, you probably think that I'm confusing nihilism and existentialism, using the words interchangeably. I used to, because I didn't understand the difference. While they are indeed very distinct, I still think they're worth mentioning side by side because in my mind existentialism is the natural progression from nihilism.

A nihilist would hold that there is no meaning to existence.

A existentialist would hold that there is no inherent meaning to existence (defying the Kantian idea of ethical maxims and such) and that it is down to you, the individual, to give it meaning.

Existentialism is then, to me, is a more mature and sophisticated iteration of the sentiment expressed by nihilists. 'There is no point in being alive' is the cold realisation that most teenagers feel as though they have uncovered for the very first time in the history of humankind while they're preoccupied with wading through some sticky angsty mire. As you grow up a bit, you realise that you will only stay as miserable and grouchy as you allow yourself to be and to use a pretty apt cliché, 'Life is what you make it.'

What nihilism looks like.

It is then that you probably pull your head of your arse. You're at least twenty years old by that point and you've wasted a good few years being absolutely intolerable to be around. You emerge from the other side of puberty wheezing, acne scarred and for all intents and purposes, an existentialist. You perhaps still know that, objectively, there is no 'point' to existence – but that its down to you to stop whining about it and carve out a point for yourself. Why are you worrying about anyone else?

So nihilism is an awkward, angsty teenager. Existentialism knows better.

Good faith and bad faith

Good and bad 'faith' here isn't used in a religious sense, but is instead a term coined by Jean-Paul Sartre to explain acting in a way that is consistent with your own beliefs and character. It's therefore highly subjective – to pick on Kant again, it's not saying act in a way that you think would hold up to the universal scrutiny of others. In fact, it's saying the polar opposite – act only in a way that holds up to the scrutiny of yourself.

This means that potentially you can, in the eyes of an existentialist, get away with some pretty grim stuff so long as your conscience doesn't bat an eyelid. If you truly believe that it's okay, then you're still acting in good faith. Then there's the argument that your conscience is simply a internalisation of what society thinks you should do – 'the policeman in your head' – and that it's not really you.

Something that Sartre particularly despises and writes about in his preface to Frantz Fanon's 'The Wretched of the Earth' is dull compliance with something you think is wrong, of being an accomplice. Whether you're literally spying on your neighbour and dobbing them in to the Gestapo or simply not rising up with the rest of the insurgents and no longer allowing your people to be oppressed, you're scum - according to Sartre. Both of those actions or inactions will be borne out of fear of physical pain inflicted by those who are threatening you, but that's not a good enough reason.

Frantz Fanon wrote about the French occupation of Algeria and their treatment of the native people there. He said that violence is the only true means of shaking off such subjugation. Sartre noted that in yet another act of bad faith that soldiers delude themselves in to being able to physically commit these atrocities in the first place by reducing the status of the natives to cattle, to animals not deserving of rights or respect.

When you do this, you breed the negative characteristics that come out of desperation like violence and thieving and then make the false assumption that these are inherent traits of these people, which perpetuates the oppression further and, even worse, lets the slavemasters feel justified in doing whatever they can to retain power. The slavemasters engineer things in such a way to make the natives turn their violent feelings on each other, and they remain too fractional and afraid to take a stand against their real enemy.

Nietzsche

Nietzsche is the stereotypical existentialist. He wrote that “God is dead”: something that initially sounds melodramatic and confrontational for the sake of being confrontational. A more optimistic reading is that the death of God and religion means that human beings are free to make their own choices, which is incredibly exciting. It gives us the freedom to find value for ourselves and not be bound up in Catholic guilt or fear of retribution for not living in a way according to outdated scripture.

Human nature is not universal for Nietzsche. Our natures are all different and therefore it follows that different people can find and follow different conceptions of ‘excellence.’ This creates room for different moralities, and Frantz Fanon’s use of violence as a means of liberation.

Facticity

Facticity is an ugly word that describes the parts of ourselves which are simply given: our gender, our age, the wealth of our family, the way we look. They’re parts of the inauthentic self – the self that society sees and wants you to be.

It’s hard not to define someone by these criteria. Tthink about who you are, and you’ll probably hear yourself thinking that you’re a 20-something student or whatever you are. It’s shorthand, of course – you can’t intimately know every person you come across, and so this is just a lazy way of getting across what this person might be like. Newspapers do this all the time; the victim in a story was a 30 year old mother of two, or a 40 year old doctor. This is because to exist, you’re ‘being in the world’ – a certain type of engagement. But you don’t have to settle simply to exist and stick to whatever society expects of you.

Transcendence

You are defined not only by your facticity but by your choices. Because you have a choice, you can re-create yourself and refuse to be defined by your past, which you can do nothing about.

The Ubermensch was supposed to be the ultimate realisation of this before the word was hijacked completely by the Nazis. The idea was that these people would overcome what has so far defined us as human, carving out their own place in the world solely according to their own will. They transcend.

The Outsider

Graham wrote a very good seminar paper on The Outsider by Albert Camus. I didn't read the book myself, but I'm wishing that I did because it seems to sum everything up rather well, or maybe that was just Graham's writing. The main character is the titular 'Outsider' – a Patrick Bateman type who seems incapable of experiencing ordinary emotions and is confused and often distressed by the pointlessness of the world and of other people's actions. A marriage proposal, the death of his mother, and even his trial for the murder of a stranger he killed for no reason at all don't seem to particularly sway him, and he remains incredibly disassociated throughout the book. He simply doesn't care about anything – to nick a quote from Graham: “Nothing matters in the world beyond the meaning we give it, and since Meursault (the Outsider) doesn’t seem to give anything meaning, nothing matters.”

He doesn't lie. He doesn't say that he regrets the murder, because he truly does not. He's not trying to conform to society or hide the fact that he's just a husk of a human being. He is a “man prepared to die for the truth,” even though that's not a necessarily positive thing in this case. He is a true existentialist because at least he is not lying to himself. He's acting in good faith.

So, to conclude, acting in good faith isn't always going to guarantee a happy ending and because existentialism pretty much destroys the idea of an afterlife and the myriad complications of doing what you think is 'the right thing' based on some arbitrary system of metaphysical reward, don't look forward to harps and pearly gates just because you think you've been noble and true to yourself.

Bit of a downer, but then again, if Hell doesn't exist either, you can do whatever you like within (your own) reason, and this is precisely the reason why existentialism and liberation can go hand in hand.