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Probing the Pudding

Writer's picture: Mishkat BhattacharyaMishkat Bhattacharya

This post is about proof - an object usually found in the pudding - and its importance. The word 'proof' comes from the Latin 'probare', which means to test, and which hopefully explains the title of the post.


Below I will first discuss the importance of proof in mathematics, keeping in mind that I am not a professional mathematician. Then I will discuss the importance of proof in physics, keeping in mind that I am a barely competent theoretical physicist.


Proof in Mathematics


Mathematical proof has existed in several forms in antiquity (in the form of plausibility arguments, etc.). The first person to exploit it at great depth and range was likely Euclid. A brief look at the Elements (proofs from which all of us learnt in high school) will convince most people that the author knew the concept of proof inside out. This included its limitations, resulting in probably one of my all time favorite quotes ("What has been asserted...").


Proofs start with axioms (assumptions) and then logically infer conclusions. Many methods of proof exist: mathematical induction, reductio ad absurdum, contraposition, computational, etc. There is apparently even a subject called proof theory where proofs are considered to be formal mathematical objects and manipulated accordingly. I know nothing about it.


In this context it might be appropriate to state that Godel's Incompleteness Theorem is often misinterpreted to imply that nothing can be proved in mathematics. That is not correct; what it says is that some things cannot be proved. This is an involved topic, and I will probably write about it in a separate post. For now, let's note that it was ironic that one of the greatest logicians of all time convinced himself that someone was trying to poison him and starved himself to death. (Or was his suspicion correct?).


Proof in Physics


I may be allowed to begin this section by confessing that mathematicians often get upset at physicists for not being rigorous enough with their mathematics, and in particular for not supplying proof for their assertions. When I was a postdoc, I actually attended a seminar in the mathematics department aimed at addressing - if not redressing - this injustice.


The snide answer to the mathematicians' complaint is that physicists do not need proof, they have experiment (or another one, perhaps more applicable to theoretical physicists: too much rigor can lead to rigor mortis). The honest answer is that physicists need proofs and use them fairly often.


There are various kinds of proofs in physics. I will discuss two types in this post.


The first type involves proofs which are essentially limited to the mathematical formalism. The most important of these are called theorems. These theorems basically help in calculating quantities of physical interest. Examples are the Parallel Axis theorem (used for determining the moments of inertia of rigid bodies), and the Quantum Regression theorem (used for determining correlation functions in quantum mechanics).


Another type of proof, though relying on mathematics, says something profound about physics. Examples in this category are the No-Cloning theorem (which shows that unknown quantum states cannot be cloned), or the Penrose-Hawking theorem (which specifies conditions for the appearance of gravitational singularities, such as those inside black holes).


Notwithstanding these examples, it generally seems difficult to prove the existence of natural laws or phenomena. For example, as far as I know, no one can mathematically prove that the sun - or anything for that matter - exists.


To conclude on a more positive note, a short list of alternative proof styles I have encountered as an academic physicist over 25 years:


Proof by intimidation - You say the same thing again, but this time in a louder voice.


Proof by erasure - You erase from the board what you wrote so quickly that no one can question it.


Proof by tautology - You say the thing is well known to those who know it well.


Proof by absenteeism - During the talk you say we can discuss this afterwards, and afterwards you make yourself scarce.


Proof by transference - You say the proof is trivial and is left as an exercise for the questioner.







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