# Are there examples of mathematical problems proven by abduction?

Proof by deduction is simple. For example:

All humans are mortal, and Bill is a human; Therefore, Bill is mortal.

However, proof by abduction differs. A famous example:

The lawn is wet. But if it rained last night, then it would be unsurprising that the lawn is wet. Therefore, by abductive reasoning, it rained last night.

There seems like a subtle difference, but it's actually huge as abduction is uncertain rather than certain.

Question 1

Are there examples of mathematical problems that have been proven by only abduction, rather than by deduction or induction?

Question 2

If there are examples of this, are they limited to problems related to probability?

Thanks.

• Abductive reasoning is a good way to get conjectures, but it can't be a method of mathematical reasoning. In particular, chains of abductive reasoning become harder and harder to justify, because "most likely" is not well-measured, so if we adjoin two "most likely" reasonings, we might not get a "most likely" result. – Thomas Andrews Oct 25 '14 at 21:32
• Another problem is that "most likely" is hard to determine in a lot of cases. For example, there are discrete number theoretic results which are true "for all natural numbers" if and only if the Reimann Hypothesis is true. We know the discrete result is true, say, up to $n<10^{3000}$. That's impressive, but does it mean the result is likely true "for all $n$?" We've verified the result still for only finitely many out of infinitely man values, yet a lot of people would intuitively see this as reason to believe it was true for all $n$. – Thomas Andrews Oct 25 '14 at 21:44
• But if we allow for other meanings of "abduction," I kidnapped a professor once and made him do my homework. – Thomas Andrews Oct 25 '14 at 21:50
• You may find amazon.com/Probability-Theory-The-Logic-Science/dp/0521592712 of interest (you can find large portions of the book freely available online). The first two chapters are an excellent introduction to plausible reasoning. – Ittay Weiss Oct 25 '14 at 22:44

As a matter of vocabulary, no, there are no proofs by abduction in (pure) mathematics; part of being properly socialized as a mathematician is reserving the word "proof" for deduction.

That's not to say that mathematicians never use plausible reasoning — it's as common as in any other field, I imagine. Plausibility arguments, often called heuristic arguments, can even be presented publicly and seen as mathematical contributions — but they simply don't constitute proof by the standards of the field.

There are a few nice examples in Terry Tao's blog post "The probabilistic heuristic justification of the ABC conjecture". These arguments are probabilistic, but not in the way I think you meant — the probability considered is not the probability that the conjecture is true.

These are very nice questions but what exactly do you mean by

"Are there examples of mathematical problems that have been proven by abduction...?"

Would you accept any abductive inference as "proof" if things are uncertain? Most likely it would only lead to justifications for further pursuit, which is its strength and purpose. Moreover, if the problem is of a hard problem of organized complexity, dealing with complex adaptive systems in general, the interconnectivity in "chains of reasoning" will become more and more evident, which can be corroborated through deduction, induction and recursion with modification. For example, one can start by asking how much connectivity one should expect of natural processes and math can be one tool that may be used to solve such problems.

Examples of the type, "the lawn is wet", tend to mask the complexity of this very important integrative quality that depends so much on sensibility/intuition.

While I am familiar with the phrase "abductive reasoning", the phrase "proof by abduction" seems like an oxymoron. Consider your example: While it is certainly true that if it rained last night it would be unsurprising that the lawn is wet, and it therefore is a reasonable inference that it probably rained last night, that is hardly the only possible explanation for why the lawn is wet. Other possibilities include:

• Dew
• An in-ground sprinkler system that waters the lawn at night (which is actually a very practical idea)

Since the idea of mathematical proof is to achieve certitude, no, there are no proofs by abduction.