Philosophy - Part 17 - Induction

Philosophy - Part 17 - Induction - by P.K.Odendaal - August 2012 

It is probable that all we know, we know from a process called induction - a very powerful mathematical and scientific tool, greatly neglected by fields like philosophy, psychology and religion. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers. In science and philosophy it is a kind of reasoning that constructs or evaluates general propositions that are derived from specific examples.

One such an example is a statement by Goldbach, called the Goldbach conjecture. In 1742 he conjectured in a letter to Euler that every even number from 4 onwards is the sum of two prime numbers like in 12=5+7. No proof of this is known other than that Tomas Oliveira e Silva proved in 2008 that it is true for all whole numbers up to 10¹⁸.

And this is the basis all our maths and science is based on. Such conjectures are known as axioms. We take as an axiom that 1+2=3, and we have a good idea that this adding process [n+(n+1)=2*n+1] is true for all possible numbers up to infinity, be cannot know for sure.

On the other hand, philosophers, like Descartes, has proposed a philosophic system much weaker than this on a conjecture which says 'I think therefore I am'. And then philosophers like Hume, Wittgenstein and to a certain extent Sartre, has thrown this reasoning into the waste paper basket.

I am quite sure there must be enough axioms that can be produced by induction that can move us forward in philosophy, psychology and religion, like they have done in mathematics and science. But what are these and where and how do we search for them? In my own conjecture I have twisted the conjecture of Descartes by saying 'I am, therefore I think' - and maybe it was not such a bad start, but we will need some serious debate over the issue.

In religion I can quote the classical story of the man who was born blind:

Joh 9:7  And said unto him, Go, wash in the pool of Siloam, (which is by interpretation, Sent.) He went his way therefore, and washed, and came seeing. v:8  The neighbours therefore, and they which before had seen him that he was blind, said, Is not this he that sat and begged? v:9  Some said, This is he: others said, He is like him: but he said, I am he. v:10  Therefore said they unto him, How were thine eyes opened? v:11  He answered and said, A man that is called Jesus made clay, and anointed mine eyes, and said unto me, Go to the pool of Siloam, and wash: and I went and washed, and I received sight. v:12  Then said they unto him, Where is he? He said, I know not. v:13  They brought to the Pharisees him that aforetime was blind. v:14  And it was the sabbath day when Jesus made the clay, and opened his eyes. v:15  Then again the Pharisees also asked him how he had received his sight. He said unto them, He put clay upon mine eyes, and I washed, and do see. v:16  Therefore said some of the Pharisees, This man is not of God, because he keepeth not the sabbath day. Others said, How can a man that is a sinner do such miracles? And there was a division among them. v:17  They say unto the blind man again, What sayest thou of him, that he hath opened thine eyes? He said, He is a prophet. v:18  But the Jews did not believe concerning him, that he had been blind, and received his sight, until they called the parents of him that had received his sight. v:19  And they asked them, saying, Is this your son, who ye say was born blind? how then doth he now see? v:20  His parents answered them and said, We know that this is our son, and that he was born blind: v:21  But by what means he now seeth, we know not; or who hath opened his eyes, we know not: he is of age; ask him: he shall speak for himself. v:22  These words spake his parents, because they feared the Jews: for the Jews had agreed already, that if any man did confess that he was Christ, he should be put out of the synagogue. v:23  Therefore said his parents, He is of age; ask him. v:24  Then again called they the man that was blind, and said unto him, Give God the praise: we know that this man is a sinner. v:25  He answered and said, Whether he be a sinner or no, I know not: one thing I know, that, whereas I was blind, now I see. v:26  Then said they to him again, What did he to thee? how opened he thine eyes? v:27  He answered them, I have told you already, and ye did not hear: wherefore would ye hear it again? will ye also be his disciples? v:28  Then they reviled him, and said, Thou art his disciple; but we are Moses' disciples. v:29  We know that God spake unto Moses: as for this fellow, we know not from whence he is. v:30  The man answered and said unto them, Why herein is a marvellous thing, that ye know not from whence he is, and yet he hath opened mine eyes. v:31  Now we know that God heareth not sinners: but if any man be a worshipper of God, and doeth his will, him he heareth.

This is a very powerful argument for taking a conjecture as an axiom. This man argues with the Pharisees, who were all followers of David Hume (so to speak) and comes to the conjecture: but if any man be a worshipper of God, and doeth his will, him he heareth.

In the process the Pharisees have their own conjectures like: Therefore said some of the Pharisees, This man is not of God, because he keepeth not the sabbath day - what a useless conjecture?

This argument of the blind man is now open for the inductive reasoning process, and has been proved for numbers up to 10¹⁸ (so to speak) - and it is still not accepted as an axiom by many people, because they do not know whether it is true for the whole domain and for all possible cases. In fact, nobody pursues this conjecture, exactly for this reason - they may find out that it is true, and then they cannot profess to be atheists or agnostics anymore.

I am not into the game of stating conjectures which may become axioms, but I know a lot of clever philosophers who have neglected this line of thought over centuries. Maybe we can start again.

The conjectures presently being pursued by scientists as axioms, are even shakier and weaker ones.

Firstly there is the conjecture which says that the Darwinian Delusion is a very close approximation of Evolution. Well - that is conjecture at its most primitive, ignorant, unproved or un-induced stage.

There is another which says that life, reality, matter and many other things end at the speed of light, whilst we know that it is not so, but we are unable, at this stage, to turn our conjectures of thought, psyche, consciousness and telepathy into more tangible axioms.

Where shall we go next? Or shall we finally accept Gödel's Incompleteness Theorems which states:

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.

Goodbye Hilbert!²

2.       Refers to David Hilbert (January 23, 1862 – February 14, 1943)


Afterthought 2 February 2013 :

Be aware that induction on which our mathematics and science is based is a trap for fools, and I will illustrate it this way :

A martian comes to our planet and find a lot of replicas of mankind. He sees the replication as a law of our planet, where every specimen of mankind is a copy of half of itself, being the left and right of two identical sides. Each person has these two identical sides with identical parts like two identical arms, two legs, two eyes, two ears, two brains, two middle fingers, two big toes ... and so on.
By induction he is allowed to extrapolate that what he cannot see in the inside of us are also identical and he confirms his postulate on a 95% probablity that we also have two hearts.

Just to bad. Science is always right. We should have two hearts.