The following was written as part of a Ethics & Integrity course held at Centrum Wiskunde & Informatica. Date: 21 November 2019. Author: Hans-Dieter Hiep

**Ethical issue**: How trustworthy is (pure) mathematical proof?

**Type of issue**: Open question, the answer is not either "yes, you can trust mathematical proof" or "no, you cannot trust mathematical proof"

**Owner of issue**: Anyone who studies mathematics in detail, or wishes to rely on the results of mathematical proof

**Who has to take action/can solve the issue**: I am responsible to any inquiry on this matter

**Why does it bother me**: Sometimes, I encounter a proof that does not elucidate a clear picture in my mind that convinces me of the truth of the statement it is supposed to prove. Sometimes, the statement being proven is itself also unclear to me. Sometimes, a whole body of mathematics seems impenetrable to me, by its sheer size and lack of clear points of entry. But, sometimes the opposite happens and I get convinced and see clearly why an argument is convincing. How can I trust my own proofs? How can I make sure my own work brings forth a sense of clarity in the minds of my audience?

Short description using reflective equilibruim.

**Principle**: "Science considers repetition in time of inter-identifiable succession-sequences of qualitative differentiation through time." In simple terms, science identifies similarities and differences between repeating experiences over time. In some sense, science ("what I know") is the most trustworthy prediction of future events I can have.

**Principle**: Trustworthiness is measured in predictability: to convincingly show knowledge is to be able to demonstrate a future result after announcing it's outcome in the past.

**Principle**: Science requires time, in the consideration of relating a sequence of past events to a sequence of future events in time, and is areligious.

**Fact**: Mathematical proof is constructed from logically expressed axioms and principles.

**Fact**: Logical application of the mathematical induction principle is not reliable in the natural sciences. It is not guaranteed that if some mathematical model explains some observations of the natural world, that all observations of the natural world are explained by that same mathematical model. In other words, if your mathematical model turns out wrong, as witnessed by the existence of a natural observation that is falsified in the model, you can always blame God (or any "thing" outside of the domain of discourse of the model) for making nature so complex to unravel.

**Fact**: In wisdom, or religious truth, logic is not reliable. Applying purely logical principles to religious truth distorts its content. A (quite violent) example of a religious thought is: imagine how a bloodstained back (of Jezus hanging on a cross) drips, like water falls from the sky, in a pool below, so that after all waves, caused by the sin of hanging this man, have disappeared / the most beautiful rainbow appears as a sign of grace of God. Personally, I imagined this for the first time, while hearing the tenor aria at the Hoogleraren Concert '19 of the Vrije Universiteit. Suppose we now apply the following (logical) critical thinking: how can *the* most beautiful rainbow appear at that moment? How would it even be possible for a rainbow to appear if blood is colored red, therefore absorbing all other colors of the rainbow? If, otherwise, a rainbow appeared in the sky, why would that be related to the death of the man hanged on the cross? Et cetera. Posing these questions and trying to answer them distorts the religious content, and shatter the timelessness of our wholly consideration.

**Intuition**: So if logic is not reliable in the natural sciences, and not reliable in religious thought: why should its application be reliable in pure mathematical proof?

**Fact**: Mathematical proof is not a physical entity, but one experienced by a human being trained in disciplined thought. "Mathematical attention is not a necessity but a phenomenon of life subject to the free will, everyone can find this out for himself by internal experience." (Van Stigt, 1990) The (physical) form in which this thought is manifest is irrelevant to the proof: it can be conveyed through diagram, English text, speech, et cetera.

**Fact**: Computer programs are a form of mathematical proof.

**Fact**: Most computer programs do not clearly state what they prove, viz. the statement being proved is lacking.

**Fact**: It is *very difficult* to me to understand the proof a computer program conveys, as it requires me to come up with a plausible statement of which the proof indeed shows its correctness.

**Fact**: Computer programs are used (in practice by billions of people) to specify computation performed by physical machines or humans pretending to be cybernetic servomechanisms.

**Fact**: Computation is the process of simplifying proof until the outcome is evidently present.

**Fact**: The outcome of the computational process is as prescribed by the statement (cf. that which is often lacking) of which its proof is manifest in the program.

**Intuition**: One cannot rely on the outcome of a process, if one can not prescribe, viz. a priori describe, what outcome is expected.

How trustworthy is (pure) mathematical proof?

In other words, how is computer (viz. computing) science a science?

I believe this is a big ethical issue, especially considering that large amounts of people depend on the outcomes of mechanical computers. In safety-critical scenarios it is essential to be able to trust the programs that specify a computational process. There are numerous examples in society, e.g. Boeing 737 max crashes costing the lives of more than 300 people and est. $4-7 billion dollar in damages and loss in income, that are caused by simplistic errors in control programs. Safety sensitive programs in use are not visible for researchers such as myself to study or understand, and this is a well-known problem.

Should we continue, as a society, to rely on computer programs of which it is unknown what outcomes they exactly prescribe? Should any scientist who does not exactly prescribe the outcome of their programs publish their programs? In other words, should a mathematician who does not state the theorem their proofs prove publish their proofs alone?

The bigger picture is this: in practice we focus *only* on the *outcome* of a program, as if it were a timeless result. It seems as if the current state of the world is built not on computer *science*, but on computer *religion*!

As a scientist, I ask: do I have any moral obligations to serve the computer religion?

Sources:

https://arxiv.org/pdf/1511.01113.pdf

https://www.youtube.com/watch?v=-Uae9_pgZzE

http://www.ppig.org/sites/ppig.org/files/2018-PPIG-29th-sharpe.pdf

https://www.youtube.com/watch?v=G21GU2BKEYQ

http://www.bach-cantatas.com/Texts/BWV245-Eng3.htm

https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence

https://www.etymonline.com/word/science