The Nature of Maths

1. The Nature of Maths Maths, Certainty, and Truth, & Maths: Invention or Discovery

2. Maths & Truth The mathematician gazed heavenward in supplication, and then intoned, “In Scotland there exists at least one field, containing at least one sheep, at least one side of which is black.” “How interesting,” observed the astronomer, “All Scottish sheep are black!” “No, no!” the physicist responded, “Some Scottish sheep are black!” An astronomer, a physicist, and a mathematician were holidaying in Scotland. Glancing from a train window, they observed a black sheep in the middle of a field.

3. Seeking Certainty Mathematicians build theorems based axioms that must be valid. Late 19th century mathematicians wanted to rebuild maths from 1st principles. The German mathematician David Hilbert led this quest for total consistency.

4. Trouble in’t mill Great strides were by mathematicians such as GottlobFrege. … Until Bertrand Russell thought up ‘Russell’s Paradox’ This showed a major flaw in the logic assumed up this point and questioned the existence of ‘completeness’ in maths.

5. Overcoming the insurmountable Three decades of trying to plug the gap. Headway appeared to be being made before Kurt Gödel provedthat maths could never be logically.

6. Gödel’s Incompleteness Theorem His theorems paraphrase to: First theorem of undecidability If axiomatic set theory is consistent, there exist theorems which can neither be proved or disproved. Second theorem of undecidability There is no constructive procedure which will prove axiomatic theorem to be consistent. (This is a simplified version of this kind of thing)

7. Huh??? Everything I say is a lie. Or: This statement does not have any proof.

8. So??? Well, nothing really... Just as Heisenberg’s uncertainty principle in physics is largely irrelevant in everyday physics. Gödel’s theorems do not invalidate any past proofs They are only relevant in the logicians world of undecidability. Few undecidable questions exist.

9. Maths, Truth, & Undecidability Gödel’s theorems have added to the beauty and complexity of maths. Axioms and proof still lead to absolute truth in the mathematical sense. The discovery that maths has its limits only reaffirms that maths is a genuinely creativity set within a formalised system of logic.

10. Invention or Discovery? • In determining whether maths invented or discovered definition is everything. • That 3 sides in a right-angled obey Pythagoras’ theorem long before Pythagoras proved (invented) it.

11. Invention or Discovery? • Hippasus‘invented’ surds with his proof of the irrationality of √2, yet every square of unit length every constructed or drawn before this had a diagonal of this length. • The Earth was the third planet from the sun for billions of years before Frege proved the existence of ‘threeness’ mathematically from 1st principles.

12. Invention or Discovery? • Newton and Leibnitz invented the calculus which is apparent any time something falls. • Bombelli invented ‘I’, the square root of -1, centuries before quantum physicists found behaviour that could only be modelled using it.

13. The real question... • It comes back to what is maths. • Much ‘maths’ has come from the observation and explanation of real-world phenomena (science). • As ‘maths’ took off in its own right, many of its discoveries have conversely been used to model and explain observations of reality.

14. My two cents: • What is the sound of one hand clapping? • If a tree falls in a forest and there is no-one around to witness it, does it make any sound? • Is maths invented or discovered?

15. Clarity: Pure maths? Applied maths? Pure maths is invented. Applied maths is observed. How to apply maths is discovered. That reality can be modelled mathematically does not mean that maths is ‘out there’.

16. Definitions please. • What is Maths – what do you mean when you use the phrase? • What is Knowledge – what do you mean by knowledge? • What is truth – what do you define as truth?

17. Over to you • This is what theory of knowledge is about: getting you guys to think & actually examine the understanding that you have acquired about/within various disciplines that claim to offer knowledge. • Maths is truth, maths is beauty, maths is the only path to understanding. • The above statement may, or may not be correct. • What do you think? ...please think... (mis)

18. Gödel’s First Theorem To every ω-consistent recursive class κ of formulae there correspond recursive class-signs r, such that neither ν Gen rNeg(ν Gen r) belongs to Flg(κ) (where ν is the free variable of r) Run away...