Nino asked:

Why did irrational numbers bother the Pythagoreans mathematical worldview?

Answer by Craig Skinner

The Pythagoreans’ worldview combined mysticism, numerology, mathematics and philosophy.

They felt that somehow numbers were the essence of all things. As Aristotle later said:

‘The Pythagoreans… devoted themselves to mathematics; they were the first to advance this study… they thought its principles were the principles of all things.’

Each number had personality and symbolic meaning eg 1(energy), 2 (fertility), 3 (time), 4 (space), 5 (the elements), 6 (resurrection) etc. Odd and even numbers were of different gender.

Each number was exact and well-defined. So, by numbers they meant the whole numbers (1, 2, 3…one million…etc) or ratios of whole numbers (hence rational numbers) such as 1/2, 4/5, 137/233 etc. All reality could be represented, and all magnitudes expressed, as whole numbers or rational numbers. Irrational numbers (not expressible as such a ratio) were an impossibility, an absurdity.

Bizarrely, Pythagoras’s own theorem forced an irrational number on them. Consider the right angled triangle with short sides each one unit. Clearly the length of the hypotenuse is square root of two (hereafter “root 2”). Hence, for the Pythagoreans, root 2 must be expressible as a ratio between two whole numbers. They couldn’t come up with which two, but in due course this might be known. Sadly, it can be easily proven that root two is not rational (proof follows in a moment). When the sect grasped this, all members were sworn to secrecy about these numbers which challenged their doctrines. Hippasus blabbed about it, and was drowned.

Geometry was clearly superior to arithmetic – root 2 could be represented exactly by a line but not by numbers. Plato’s model for his views on forms and universals was ideal geometrical objects, and, famously, the sign over his Academy said ‘Let no one ignorant of geometry enter this house.’

Of course things have moved on since the Greeks, with successive acceptance (sometimes resisted by many mathematicians) of irrational numbers, negative numbers, zero, transcendental numbers, imaginary numbers and nimbers.

Proof that root 2 is irrational:

Assume it is expressible as a ratio of two whole numbers. Let these, reduced to their lowest common form by division, be m and n.

It follows that m and n are not both even (if they were we could divide both by 2 till one or both is odd). So one or both of m and n must be odd.

So, root 2 = m/n

(squaring both sides), 2 = m²/n²

Hence m² = 2n²

Hence m² is even

Hence m is even

Hence m = 2p

Hence m² = (2p)² = 4p²

Hence 2n² (= m²) = 4p²

Hence n² = 2p²

Hence n² is even

Hence n is even

Contradiction (m and n can’t both be even)

Hence assumption that root 2 = m/n is incorrect (reductio ad absurdum).

Angie asked:

What is digging beyond the obvious? I need further explanation on
this matter. Thanks.

Answer by Helier Robinson

The obvious is not necessarily true. For example, if you have a prejudice then you unconsciously remember all the evidence in favour of it and forget all the evidence against it, thereby making the prejudice obviously true. Thus to a fundamentalist Christian it is obvious that everything in the Bible is true, hence the theory of evolution is obviously false. Or you might be thinking stereo-typically: you might have met a redhead who was terrifying, so that it is obvious that all redheads are evil. Or you might be superstitious, so that it is obvious that crossing your fingers and touching wood avert bad luck. As well, what we call common sense is obviously true but was sometimes wrong in the past: it used to be a matter of common sense that the Earth was flat and at the centre of the Universe, and that human beings were specially created; who is to say for sure that all present day common sense is true? Furthermore, there is a peculiar feature about belief: every belief carries a piggy-back belief, to the effect that the belief it rides is true, and this makes the belief obviously true. You may have noticed that while most other people have beliefs that you are sure are false, your own beliefs are all obviously true.

Digging beyond the obvious is what people do if they are really interested in finding the truth, and do not know if what is obvious to them is true or not. The most successful way to find truth is to study science; the next most successful way is to philosophise, or if this is too difficult, to study philosophy. Not easy, but sometimes very exciting.

Phil asked:

How can one best read philosophy so as to increase one’s understanding of it?

Answer by Shaun Williamson

I think one thing you need to do at some stage is to read a history of philosophy. This will give you an overview of how philosophy developed. Bertrand Russell’s one volume history of Western Philosophy is available for free over the internet (at http://www.gutenberg.org). The multi volume history by F. Copleston is also worth reading.

You should keep in mind that any history will reflect the bias of its author but both of these will give a useful overview of the history and scope of philosophy and may give you ideas for further reading.

In general you should keep in mind that philosophy is a difficult subject, as difficult as nuclear physics or higher mathematics so don’t expect to understand any book the first time you read it. You need persistence and you have to be prepared for your brain to hurt.

Amol asked:

What is reality?

Answer by Helier Robinson

There are three usual definitions of reality.

The first is that reality is all that we perceive around us that is potentially universally public. These three conditions apply for the reason that some of what we perceive around us is illusory. You might be wearing coloured sun-glasses, for example, so that the moon appears to be blue. But you quickly discover that this blue moon is private to you. Such private perceptions are illusions, and unreal. They are mis-representations of reality. Another example is that everything visible appears smaller with distance. Imagine a straight road with telephone poles along it: with distance, the road gets narrower, the poles get shorter, and the poles get closer together; visible space shrinks with distance in all three dimensions. Artists have to use a special kind of geometry, projective geometry, to draw visible space correctly. But this shrinkage, although public, is illusory. Suppose that you are on a straight road on the way to meet your friend and you see her in the distance; you text her that you can see her in the distance and that the road appears to be very narrow where she is; and she will reply that it is wide where she is but narrow where are you are. So this illusion, although public, is not universally public. But we cannot achieve universal publicity, so we say that the real is what we perceive around us that is potentially universally public. This kind of reality might be called empirical reality, since the empirical is all that we know through perception.

The second definition of reality is that it is all that exists independently of human perception, or even of human consciousness. That is, it exists regardless of whether anyone perceives it or not. Included in this reality are things which cannot be perceived but for which we have good evidence: electrons, black holes, minds other that one’s own, empirical objects when no one is perceiving them, etc. This kind of reality might be called theoretical reality, since the theoretical is non-empirical.

The third definition of reality is that it is all that makes true propositions and beliefs, true. It is the ground of truth, in other words.

Paul asked:

What is beauty?

Answer by Tony Fahey

Whilst in philosophy in particular, and academic discourse in general, it is advisable to avoid sweeping statements, I believe it is fair to say that the issue of defining beauty is one that has occupied the mind, not just of scholars, but of many people over several millennia. For example, for Plato, an ideal from of beauty is not found in the natural world: in individual things such as objets d’art, people, animals and so on, but in the Realm of Ideal Forms which it shares with other forms such as Justice. Plato acknowledged that everything that belongs in the material world is made of substances that time will eventually erode. However, he also held that everything is made after a timeless mould that is eternal and immutable; that is, everything in the physical world is but an inferior copy of an ideal form which exists in, which for him, is the ‘real’ world.

According to Plato then, all that we perceive in the material world are but poor imitations of these original forms. Against this, Aristotle argues that there is no such as an absolute or ideal form of beauty. Things in the material world are real. Nature, he says, is the real world; things that are in the human soul/mind are not, as Plato holds, a priori concepts, but concepts and ideas gained by empirical experience. That which we perceive as beautiful, he argues, is not some poor imitation of that which has its archetype in a metaphysical realm, rather it is a representation of a quality intrinsic to the thing itself.

However, rather than continuing with a detailed and long-winded history of the issue of the question of beauty over the centuries, let me give you my own understanding of this concept as I have come to see it.

In Philosophy the rubric under which the issue of beauty is discussed is aesthetics. The term aesthetics derives from the Greek word aisthanomai, which means to perceive, to feel and it is in this ancient term that we find the essence of the meaning of that which we know as ‘beauty’. That is, it is the appreciation by the mind of the quality we recognize as beautiful in phenomena (i.e. things in the world outside the mind) transmitted to the mind through the senses. The question, of course, that arises from this description is: ‘what is it in the mind that allows it to make this judgment call?’

In the issue of the relationship between mind and phenomena, from Kant we learn that ‘although all knowledge begins in experience, it by no means follows that all arises out of it’. What Kant argues is that before we experience things in themselves (phenomena), there already exists, within the mind, a certain a priori framework that allows us to give meaning to that which we experience through the senses. For Kant, this framework is made up of the intuitions space and time and the law of cause and effect. Now, it seems to me that in the much the same way that Kant makes the case for space and time, and the law of causality privileging the mind to put order on that which the mind experiences, so too can a case be made that there also exists within the mind (let us call it) a property that privileges it to make a judgment call on that which it experiences or perceives as beauty. That property is what I call the instinct of equilibrity.

Let me explain:

Human consciousness privileges us with an awareness of our existence. Intentionality, as a feature of consciousness, privileges us with the wherewithal to contemplate affairs of the world. However, in order to put order on that which we experience, as well as space and time and the law of cause and effect, nature has furnished the mind/brain with another, equally important, feature which I call the instinct of equilibrity: that is, an innate sense of equilibrium which is essential in the making of judgment calls necessary not only for our safety and development, but also for our appreciation of that quality in things which we have come to describe as beautiful. It is in virtue of this feature that the mind rises above the prosaic or mundane, to focus on that special indefinable quality in things that makes them worth experiencing simply for the enjoyment or pleasure from that which they are in themselves. It is in virtue of this feature that we develop our sense of justice, goodness, truth, and beauty – qualities which, whilst perhaps indefinable in themselves, are essential in establishing an environment in which human beings can live, prosper, grow, and even come to appreciate that which find aesthetically pleasing or beautiful.

It seems to me that it is this instinct, this sense of balance or proportionality, that was in Keats’ mind when he said that ‘Beauty is truth, truth beauty! – that is all/Ye know and all ye need to know’. That is, at its most refined, beauty is truth, and truth is beauty.

Wilona asked:

What is life?

Answer by Helier Robinson

The best definition that I know of was given by Erwin Schrodinger, the physicist of wave equation fame, who suggested that a living organism is a system of very high negative entropy in dynamic equilibrium. Entropy is a measure of disorder, and negative entropy is a measure of order. Dynamic equilibrium is equilibrium through change. For example, a car driving along a road is kept in dynamic equilibrium, relative to the legal side of the road to be driven on, by the driver; if the driver falls asleep then the equilibrium collapses. According to the second law of thermodynamics entropy can increase but not decrease, so that the opposite applies to negative entropy: it can decrease, but not increase. But living things manage to avoid this by feeding on negative entropy: sunlight, carbon dioxide, and oxygen for plants, and plants or other animals for animals. (The fate of every living thing is to live for a while and then be eaten by something else, with the exception of those that die in forest fires, volcanic eruptions, etc., and humans who do like the idea of being eaten and so get cremated or buried.) Your negative entropy is lower if you are tired or ill, and its dynamic equilibrium collapses when you die. See Schrodinger, What is Life? Cambridge University Press, 1948.