Santy asked:

Were the Ancient Greeks correct in thinking that mathematics is an absolute truth?

Answer by Helier Robinson

Yes, definitely. But you have to understand that the truth that is absolute is mathematical truth, not empirical truth. All the mathematical truths that the ancient Greeks discovered are as true today as they were then — Pythagoras’ Theorem, for example. It does not matter what language a mathematics book might be written in, the mathematical statements proved in it will be true in any other language.

It does not matter what mathematician discovers a new mathematical truth, if he proves it then all other mathematicians will agree that it is true. It may be modified under special circumstances, but is otherwise absolutely true: for example, a right-angled equiangular triangle is impossible in Euclidean geometry, and that is absolutely true; but it is possible in spherical geometry. In spherical geometry, which is geometry on the surface of a sphere, the shortest distance between two points is a great circle; lines of longitude are great circles, as is the equator. One equilateral triangle is formed by the lines of zero longitude, longitude 90 west, and the equator — and this is a right-angled equiangular triangle, possible in this geometry because all triangles in this geometry have their interior angles sum to more than 180 degrees, unlike Euclidean triangles which sum to exactly 180 degrees.

The history of logic is interesting in this context. Formal logic was invented by Aristotle and developed by the medieval philosophers. It is a logic of classes (hence our word ‘classification’). Modern symbolic logic is a logic of truth-functions and, in quantificational logic, a logic of sets — and sets, basically, are classes. All of this is derivative from the logic used by mathematicians, which has never been formalised but which includes sets and functions and valid argument forms such as modus ponens and modus tollens and all the other valid argument forms of formal logic — as well as purely mathematical argument forms such as mathematical induction, and concepts not appearing in formal logic, such as numbers, shapes, equations, and algebraic and topological structures. So if you are interested in reasoning, study mathematics, not logic.

Robert asked:

Why does an infinite regress have to be terminated?

Answer by Geoffrey Klempner

I’m guessing that what you mean by your question is what is so bad about a regress that is NOT terminated, and hence infinite.

An infinite regress which is considered bad is sometimes called a ‘vicious regress’.

An example of an argument that uses the idea of an infinite regress is the Cosmological argument for the existence of God. If we trace chains of cause and effect back far enough we either get to a Big Bang (the beginning of the universe) or the causes and effects go back for ever.

If the series terminates with the Big Bang, then that would be an event without a cause, contrary to the belief that every event has a cause. If the series doesn’t terminate but just keeps going back further and further in time, that would be an infinite regress.

But why is the regress vicious?

First, I need to explain the difference between an infinite regress — for example, a regress of causes and effects, or of explanations — and an infinite series.

In mathematics, there are examples of an infinite series that is terminated at one end and and not the other, e.g. the series 0, 1, 2, 3… which terminates at 0, or the series 0, -1, -2, -3… which also terminates at 0. Or you can have a series that is infinite in both directions, e.g. … -3, -2, -1, 0, 1, 2, 3… . It doesn’t terminate anywhere. Nothing bad or vicious about that at all.

What is infinity? In mathematical terms,

“An infinite set is a set whose members can be put into a 1-1 correlation with a proper subset of itself.”

What that means in layman’s terms is that if you remove half of an infinite series, you still have the same ‘number’ of items. E.g. 0, 1, 2, 3… can paired off with 0, 2, 4, 6… That’s just one of the weird properties of the infinite.

In mathematician David Hilbert’s story of the Grand Hotel (cited by George Gamov in his book One Two Three Infinity, 1947 — the inspiration for 123infinity.com) there is still room for an infinite number of new guests even though each one of the infinitely many rooms is occupied — the old guests are simply asked to move to the room which was double the number of the room they were in before, leaving an infinite number of empty rooms. Job done.

Back to the cosmological argument. We have a problem — or so it is alleged. Identifying the cause of an event is supposed to supply a reason or explanation why that event happened. The event occurred because of the cause. Going back in time, that cause occurred because of another cause and so on.

Here’s an example that illustrates the problem. In an empty field, you discover a magnificent chandelier on a chain going up into the sky and through the clouds. Each link in the chain holds the link immediately below it. Then you are told that there’s no ‘ceiling’ the chain is attached to, no first link. The chain just goes up for ever to infinity. But if there isn’t a first link then what’s holding the entire chain up?!

The problem of a vicious regress, in short, is a problem of dependency. A required explanation is indefinitely deferred. An indefinitely deferred explanation is no explanation at all.

Theists hold that God from a vantage point outside time is responsible for the entire physical series of causes and effects — the entire ‘chain’ — which is either finite or infinite, depending on whether or not the Big Bang theory is true. God is the ultimate, non-temporal ’cause’. If the series of ‘links in the chain’ is infinite, God can still be there outside of space and time to ‘hold it up’.

Is that a good argument? If you don’t believe in God, then the alternative is to say that there is no ‘ultimate cause’. Things just are the way they are rather than some other way. The existence of the universe is contingent, but it is not contingent ON anything. It just IS.

Say that if you like, but there is still a question, a problem. Maybe a singular event without a cause, or alternatively an infinite regress of effects and prior causes is bearable, but there is some discomfort there that indicates that there is something that we do not fully understand about the nature of our world.

That’s not a problem for physics but for metaphysics.

Mason asked:

Why isn’t death something one suffers from?

Helena asked:

Is it for the best that people eventually die?

Answer by Geoffrey Klempner

I’m going to answer Mason’s and Helena’s questions together because they both concern the problem of our fear of death. (See my article Is it rational to fear death? and my YouTube video What is death?)

The short and boring answer to Mason’s question comes from from the Greek philosopher Epicurus: ‘Death is nothing to us … It does not concern either the living or the dead, since for the former it is not, and the latter are no more.’ To ‘suffer’ from something (e.g. a bee sting, an amputation) you need to be alive. Death as such, as opposed to the process — possibly long and agonizing — of dying isn’t something you suffer because at the very point where death begins, ‘you’ are no more.

So far as the danger that Malthus predicted of the world population increasing at a geometric rate is concerned, it is very definitely a good thing that people eventually die — good for everyone else on the planet! But how can it be ‘good’ for you? Wouldn’t you prefer to live forever if you had the choice? That’s Helena’s question.

Thomas Nagel has written a very good essay on this in his book Mortal Questions, 1979: ‘Death’ (posted at stoa.org.uk).

One question that Nagel considers that has gripped me is the seeming paradox of the unappealingness of the thought that I might just go on for ever and ever, and the desire, at any given time, to go on living. Let’s say I’m convinced by the argument that life would get extremely boring and repetitive after, say, a thousand years. I would not want to live as long as that. And yet, as each new day begins, I sincerely hope that I do not die today.

In practice, as the centuries wear on, and you become increasingly aware of your miserable finitude and the limits of what you are able to achieve given your limited powers, the depression would increase to the point where you felt impelled to kill yourself. The maths of this situation (which is parallel to a thunder clap, or tearing a piece of paper) has been well researched: it’s called catastrophe theory. The point is that when you do finally turn the gun on yourself, you don’t do so for a ‘reason’ (which has existed possibly for hundreds of years). You just finally snap.

But what about this claim — possibly contentious — that even granted immortality, I am a limited being with limited possibilities? Surely (this is a point that has been made to me) if we are going into the realms of magic, and immortality is a magical notion (you even survive the death of the universe and the birth of a new universe!) then why couldn’t you magically acquire greater and greater intellectual powers, so that you were fully able to make use of your indefinitely extended life span?

Imagine what you like. The question is, are you imagining YOU? Are you still there, or has something over time taken your place (and perhaps fondly preserves your memories, as one keeps faded photographs in a family album). The god-lie entity that exists now, after hundreds of thousands of years, isn’t you. ‘It’ merely remembers another being’s ‘memories’, the being that you were.

Ngole asked:

How can human beings marries animals or marry fellow man?

Answer by Geoffrey Klempner

An intriguing question. First, we have to ask, What is marriage for?

Two individuals take a solemn vow to one another to be partners throughout their lives — ‘until death do us part’.

Marriage is typically consummated by sexual relations, but it need not be. According to www.gov.uk, ‘You can annul a marriage if… it wasn’t consummated — you haven’t had sex with the person you married since the wedding (doesn’t apply for same sex couples).’

The parenthetical qualification seems to introduce two concepts of ‘marriage’, which may be acceptable in law in terms of the legal consequences of marriage for inheritance or tax purposes, but seems wrong in principle. The idea that consummation is essential to ‘real’ marriage (i.e. between a man and a woman) is archaic and has no place in a modern legal system.

The exception also gives the lie to the claim that gay couples in the UK can now call themselves ‘married’. They can use the word. The legal benefits are sorted out. Yet the law still states, archaically, that there are two kinds of ‘marriage’, not one, those that can and those that can’t be annulled on the grounds of non-consummation.

Marriage may be for the sake of having children, or at least conducted with the prospect of children, but again this is not necessary, for various reasons which we need not go into. In the UK, you can’t annul a marriage on grounds of infertility, whether or not this was known in advance.

So why is it necessary that one marries another human being? Let’s say that I am a hunter, and I am in love with my seven year old mare whom I have owned since she was a yearling. We’ve been together through thick and thin. I want to get married and I think she would be happy with the idea — if only she could talk and understood my marriage proposal.

But there’s the rub. Marriage is a vow undertaken by two individuals to one another. You can’t make a vow if you lack the mental capacity to grasp what that means. My mare will never comprehend my feelings for her, even though I know in some way she appreciates the loving way in which I groom and feed her.

Of course this is ridiculous. But I sense, from the tone of your question, that you think that a man marrying another man is in the same category as a man marrying a horse.

If that is the case, poor you.

Colin asked:

What is the definition of ‘definition’?

Answer by Geoffrey Klempner

Ah, what a great question! I don’t recall anyone asking this (going back to when Ask a Philosopher was launched in 1999). However, earlier this year in an answer to a question on the difference between ‘real’ and ‘nominal’ definitions, Hubertus Fremerey states:

“My definition of ‘definition’ would be: A method of bringing some order into the boundless chaos of experiences — sensual and intellectual.”

This is a nice idea, but to my ear falls short of the requirements of a definition. Fremerey goes on to say:

“The whole concept of ‘real definition’ is a misnomer, and even ‘nominal definition’ is. What you have as primary givens are experiences, and then you first attach labels to them and if needed you re-define the (sensual or rational) objects in the context of a theory.”

This is Fremerey’s ‘take’ on the question. A take isn’t a definition, although the way you take something can be relevant to formulating a definition. A case could be made that when analytic philosophers ‘define’ concepts (the concept of a person, or causation, or event, or etc.) what they are really doing is offering takes or theories. What they are looking for is an insightful view of the concept in question and how it fits in to our conceptual scheme.

A notorious case would be the fruitless attempt to define ‘knowledge’, with ever more elaborate sets of conditions, designed to cope with every possible counterexample. A theory of knowledge is what one is after, but a theory doesn’t necessarily require that you give a set of conditions that uniquely identify the concept in question.

So let’s narrow the question and concentrate on definitions rather than theories. A definition of a term should give you sufficient information to be able to use that term successfully and correctly, provided only that you understand the terms used in the definition. The Oxford English Dictionary (‘on Historical Principles’) is a model of this approach, which offers examples of the use of the word in question, especially early or first uses, as well as a sketch of its etymology.

A popular question in English-speaking philosophy in the 50s or 60s would have been, ‘What is the difference between a philosophical analysis and a dictionary definition?’ Now that I know how, e.g., the English word ‘person’ is used in normal conversation, its derivation from the Latin ‘persona’, what more do I need to grasp the concept of a person?

Let’s stick with this example. Suppose that the predicted breakthroughs in AI come to pass, and the first intelligent creatures with artificial brains roll of the production line. Are they persons? Whom should you ask, a philosopher or a lexicographer?

Imagine a future dystopian society where AI creatures are exploited and abused because they are not regarded as ‘persons’ by the general public despite the objections of philosophers. Or, alternatively, a society which happily embraces these mechanical beings into the ‘human race’ despite the objections of philosophers. A lexicographer is interested in how a thing is regarded, without questioning too hard the basis for the belief in question. The philosopher is the one who asks, ‘But is it really?’

As stated above, the philosopher may only be able to offer a take, not a definition. But a take can be sufficient to defeat a false definition.

What is a definition of ‘definition’? There is more than one kind of definition. I have given two, for the sake of contrast, but there are probably more (consider, e.g. the use of definition in mathematics). If there are two kinds of definition, then there are two (or possibly four!) definitions of ‘definition’, and if there are three, then etc. The point, however, is to decide what kind of definition we (as philosophers) are primarily interested in.

John Smith asked:

If abstract objects are non-causal and human brains are just physical objects how are we able to know about things like numbers, sets, ethical properties, etc.?

Suppose at t1 I do not know about some abstract object x, then at t2 I come to know about x through intuition or whatever method for acquiring non-inferential knowledge of abstract objects the anti-nominalist has in mind. How exactly is X not causal, if it can bring about changes in the physical world; e.g. in my brain?

Answer by Geoffrey Klempner

A meaty question in metaphysics. I could see this in quotes followed by ‘Discuss’ on a Sheffield University BA Metaphysics exam paper. (I once gave a Metaphysics course at Sheffield, although this would not an exam question I would have personally chosen.)

By a remarkable coincidence, you share your name with the philosopher and theologian John Smith (1618-1652) who along with Henry More, Ralph Cudworth, and Benjamin Whichcote formed a group of Plotinus enthusiasts who were known as the ‘Cambridge Platonists’. This is relevant to your question, which concerns the platonist-nominalist debate, still in various guises a live topic among contemporary analytic philosophers.

One thing often lacking, unfortunately, in the analytic approach is a sense of history, so I am going to sketch briefly what it means to be a Platonist — a true Platonist rather than someone who merely dislikes the ‘nominalist’ idea that the physical world of concrete particulars is all that really exists, everything else being a more or less artificial ‘construction’.

When we talk about ‘numbers, sets and ethical properties’ we are not really referring to any entity, according to the nominalist. There is nothing ‘out there’ beyond the physical realm for our words to ‘name’. These are just words that we use according to rules.

As you correctly note, causality is the crucial notion. But how do you know that causality necessarily involves physical objects or events? That’s something that Plato would have strongly denied.

In a seminal article ‘Causation in Perception’ (in ‘Freedom and Resentment’ 1974), P.F. Strawson makes the case for a necessary link between perception and causation which does not involve the traditional, and questionable Lockean idea of a causal link between public ‘things’ in the world and private ‘ideas’ in the mind. It is essential to perception, Strawson argued, that there are ways and means by which we come to perceive objects, that, for example, perception requires light, and that it can be distorted or obstructed by things getting in the way.

Plato, in his dialogue Phaedo makes the remarkable claim that the soul is ‘akin’ to the Forms, as one of his arguments for the existence of a non-physical soul. We could not have, e.g. the idea of equality (the Form of Equals) if our soul was not of a similar nature as the Forms. There has to be a causal link there, albeit a non-physical kind of causality. In the Republic, he tells a story about how the philosopher comes to know the Form of the Good, which like the sun in the physical world provides the necessary illumination by means of which the other Forms are perceived by the soul.

Goodness plays a crucial role here, because ‘virtue’ as Plato conceives it just is aligning our souls with the order of the universe, which necessarily involves right action because you could not know, e.g. the Form of Justice without being motivated to act justly. Wrong action is a form of misalignment of the soul, obstructing intellectual perception of the Forms.

Contemporary ‘platonists’ so-called (with a small ‘p’) won’t have any of that. They may or may not think that ‘ethical properties’ are real (you can be a platonist about mathematics and a subjectivist or nihilist about ethics). Gottlob Frege (whom I mentioned in a previous answer) strongly believed that numbers ‘exist’ in their own right, despite giving, in his brilliant Foundations of Arithmetic (1984), an effective recipe for parsing away reference to numbers in favour of the logic of quantification.

So what was Frege’s story about ‘intuition’ or ‘perception’? He was a mathematician, and like many mathematicians could testify to the powerful experience of ‘seeing’ logical and numerical relationships. Unlike computers, human beings rely on their capacity for ‘vision’. Is this just metaphor? What is its cash value?

I don’t have any problem with saying that numbers are real, nor do I feel the need to ‘define’ them in terms of some other abstract objects such as sets. A number is what it is and not another thing (if you go down the definitional route you are faced with more or less arbitrary choices, so why bother?).

Are tables and chairs real? They are and they’re not. Lacking Locke’s ‘microscopical eyes’ we are forced see aggregations of atoms and molecules as single ‘objects’, which they are not ‘really’. But then neither are atoms, etc.

Why not just say, that anything you can talk about has the kind of ‘reality’ that is appropriate for the thing in question. Corresponding to this, there are any number of different notions of causality. We chunk things in different ways according to the topic. Causality and perception are everywhere and at every level (you can ‘perceive’ with the aid of an electron microscope). There is such a thing as intellectual perception, even if it does not have the metaphysical baggage that traditional Platonists or Neoplatonists gave it.

Is there no solid ground for our conceptual scheme? Why does there need to be? When Ronnie O’Sullivan strikes the white snooker ball with his cue so as to pocket the red into the center pocket, then ricochet off the black to come back into the perfect position for the next shot, his action has ’caused’ this to happen, through his exquisitely refined judgement that has no correlate in the mechanics of rebounding snooker balls. You will look in vain for an explanation on the level of physics — the ‘causation’ isn’t there, or rather there are too many ’causes’ and ‘effects’ but none of them are relevant in explaining what just took place at the snooker table.