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Comment by msla | Hacker News Reader

Mathematics isn't a science because it's deductive: It has absolute truths, and ways to derive new truths from those truths, recursively. Science, on the other hand, is empirical and thus inductive, and thus has no access to truth (problem of induction) and can only improve models based on observation.

Science uses mathematics to work out the implications of precise prediction-generators (often called laws) and check them for logical consistency, but mathematics itself has nothing to say about any world outside of the one generated by its axioms.

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Mathematics may be defined deductively, but practically speaking I don't believe it is always or even usually done that way. People came up with axioms for set theory after inventing set theory first.

There is an intuition to mathematics, but it is definitely deductive because proofs follow from definitions, as opposed to definitions being generated from specific proofs.

Definitions do sometimes end up being justified by their consequences though, eg 0^0 is just defined to be 1 because it results in nicer equations. For that matter, you can sometimes do mathematics with only rough definitions and then make it well-defined later- umbral calculus "worked" before anyone actually set it on a solid mathematical grounding:

https://en.wikipedia.org/wiki/Umbral_calculus

(you can see in that Wikipedia article the umbral calculus was originally invented and used because it seemed to work, long before anyone found a set of definitions to justify it to modern standards)

The axiom of choice is accepted because it's useful and required for certain results to hold that mathematicians aren't willing to give up, etc. If ZFC were to be somehow found to be inconsistent it would probably be patched up by altering the axioms rather than wholesale tossing out all the theorems.

I guess what I mean is that mathematics uses deduction in proofs but that's really not the end (or the beginning) of the story.

Nothing you mention makes mathematics anything other than deductive.

The article uses Turing machines as an example of computer science so the author clearly has a definition of science that includes deductive mathematics.

Turing machines are part of the prediction-generating framework of CS, like how the mathematical models in the Standard Model are part of the prediction-generating framework of modern physics: They're both an idealized representation of something which we think models something useful in the physical world in a way that allows us to think more clearly about some aspects of that thing without getting caught up in distractions. As alex_smart says, the Church-Turing thesis connects the idealized Turing machine to modern computers in a way somewhat similar to how the Standard Model connects the U(1) group to the study of electromagnetism.

I don't quite agree. The essential piece that transforms the study of Turing machines from mathematics into science is the Church-Turing thesis. And the Church-Turng thesis that is not deductive in any sense of the word.

I agree the Church-Turing thesis is why we care about Turing machines, but to say that the study of Turing machines is a science because of the Church-Turing thesis is like saying “it’s a science because it has applications.”

Anyway, bringing this back to QC. It is believed that algorithms exist that can be computed by a QC that cannot feasibly be done by a classical computer, so QC also meets the ”deductive mathematical definition with applicable characteristics” bar as well.

The Church-Turing thesis is what makes makes a result about Turning machines a theory about the empirical world rather than mere facts about a simplistic mathematical model of computation. So, the point is more like - "it's a science because it is an empirical observation about the natural world".

No, computer science is still deductive even with the C-T thesis. CS is definitely not empirical. If I think of an algorithm, I can explain its properties mathematically without experimentation.

I am not sure what it is about my way of stating things that is causing such severe comprehension problems for you (repeatedly). I never said that Computer Science is not deductive. In fact, the reason we even have a question over Computer Science being a science or not is that most of the things computer scientists do in practice are in fact deductive results about mathematical models. To be called a science, a field of study does however need to have a connection to the empirical world - and for computer science, C-T thesis is that connection.

Anyway, let me be the one bringing the discussion back to quantum computing this time. I don't think anybody in the world who could argue that study of entanglement - think Bell's theorem and CHSH inequality - isn't science. In fact, the 2022 Noble Prize for Physics was awarded to people who made foundational contributions on that field. I see Quantum Computing as a direct spiritual successor to that study.

I don't think I'm having trouble understanding you at all. Rather, your definition of science is inconsistent and thus not comprehensible. To understand why, let's assume that the C-T hypothesis was never hypothesized, but Turing still described his mathematical formulation of the Turing machine. Mathematicians then went on to use that definition to do mathematical research not tied to the empirical world. According to you, those mathematicians are not doing science. Now let's assume that someone formulated the C-T hypothesis. Suddenly, all that work that was previously "not science" is suddenly "science." So you can see how your definition, at best, can only classify things as "science" or "maybe science" because ties to the empirical world can always be discovered later.

A proper definition of science would be able to consider a study X and classify it as science or not science using only intrinsic properties of the study, not an extrinsic property like a known connection to the empirical world.

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