And, as Einstein almost said; “The universe is not just curved; it’s BENT”.

]]>Twenty-some years later, my kids, taking the same Intro To Science-type courses, also covered such topics, but never down to the level of detail of rolling the balls down the ramps with a stopwatch themselves.

They had to fit in important concepts such as, how do women and minorities feel about acceleration, and there are only so many days in a school year.

]]>But was it ever any different?

Of course not. But people of the same age who went through even the most basic schooling in 1880 or 1920 or 1950 would understand the concept implicitly, or “intuitively”, because back then they were taught how to *think* – not how to *feel*. And, it is not about reality and empirical evidence either – it is about how one is supposed to interpret it.

Ferox: What percentage of millenials do you suppose could explain the concept of falsifiability? (…) not anymore.

But was it ever any different? In the past most folk went into a productive stage of life rather than attending something calling itself a ‘university’. So Generation Snowflake have less excuse not to grasp falsification, for it is the very key to understanding reality in my view (but that is just a theory for which I have formed a critical preference 😛 ), but I suspect if you asked a random sampling of people on the streets of New York or London or Paris, in 1880 or 1920 or 1950, you would get an equal number of dubious replies.

]]>In what follows, I need to tread very carefully. This because I am not a mathematician, let alone a theoretical mathematician. Nor am I a philosopher.

I like Alisa’s challenge at October 19, 2016 at 2:47 pm:

how many know the difference between an axiom and a theorem?

This is a valid doubt about the thought processes of many people.

But things, IMHO, get worse.

Most theorems require a mathematical basis, for example arithmetic, logic, algebra. This to go from axiom to deduction; also from a mix of deductions (possibly including axioms) to further deductions.

But it is (again IMHO) an axiom that (for example and most importantly) arithmetic works – everywhere it looks as if it should work (which is pretty much everywhere). Likewise for all other parts of mathematics.

These axioms, like many other axioms, are actually supported by real-world evidence: they have been subject to the test of falsifiability – and they have passed that test posed by every piece of available evidence. So we believe they are basic (so axioms).

We might find it wise to avoid such complexities, for the vast majority of students at the primary and secondary schooling levels. Thus keeping to the simpler explanation of differentiation of axioms and theorems – and only expecting the better fraction of pupils to grasp that concept beyond its simplest explanation.

We then get to the issue of rote learning of ‘scientific facts’ versus ‘understanding’ of the underlying science – perhaps going back to first principles.

I have always been very keen on checking things back to first principles – it shows the lie in a great many seemingly sophisticated arguments. But even so, life is too short to know the path from first principles to everything we do know (that is ‘true’). And a major reason we have evolved beyond the higher apes is because we have (much greater) ability to learn from generation to generation – by relying on the diligence of our (societal) ancestors rather than doing everything ourselves from scratch.

We need (and will need forever) some people for every scientific fact: that know thoroughly the route from first principles – but we each individually do not need that ‘first principle’ information ourselves to use the conclusions (deductions from axioms and evidence).

Best regards

]]>Could they not go to learn at the feet of a shaman, who obviously would not take money to pass on his wisdom…

Insinuating that African shamans are fools who give away their knowledge (or their services) – that’s *racist*! They are professionals who charge substantial fees.

knowing e.g. that the axiom of choice was an axiom, and then looking at theorems that depended on it, was part of my university curriculum.

I was using ‘school’ as a general term – i.e. to include university, or any other schooling, for that matter. FWIW though, axioms, theorems, and the requirement to prove the latter based on the former was very much part of my high-school curriculum – but that (rather ironically) was behind the Iron Curtain.

That said, a great many people today will get the difference between an axiom and a theorem after a trivial introduction

Of course – problem is, it rarely happens any more.

I agree Bobby, but another problem is that fewer and fewer young people turn (or are turned) to STEM education to begin with.

]]>The problem is that the non-STEM fields – the sociologies, the poli-sci’s, the diversity fields – have all decided to cloak themselves in the mantle of “science”, too, because their imperfect understanding of what “science” means gives them leave to label their half-formed thoughts and desires as axioms, requiring no proof because they are simply true.

And if they can label their own fields as “science”, we should not be surprised that science holds no value for them.

]]>By contrast (answering the point of Alisa, October 19, 2016 at 2:47 pm), knowing e.g. that the axiom of choice was an axiom, and then looking at theorems that depended on it, was part of my university curriculum. Axiom and theorem were not *wholly* absent from my school but I think a pupil could have missed them or forgotten them very easily.

That said, a great many people today will get the difference between an axiom and a theorem after a trivial introduction – including I think some with uncolonised minds though probably not those with decolonized minds.

]]>Thomas Reid (natural scientist as well as philosopher) was correct.

And the relativist philosophers (also “white males” by the way) were and are wrong.

]]>So while I think I get your point and agree with it, I would put it rather differently, by asking: how many know the difference between an axiom and a theorem?

]]>How many of them do you suppose could articulate that empirical science doesn’t deal in Truth (that’s philosophy), nor even in Proof (that’s math), but rather in an iterative process of observation, hypothesis, experiment, and revision? That, like a great game of King of the Hill, scientific theories are never proven, never declared True, but are always required to defend themselves against any new test that is devised for them?

How many of them understand that process to be the strength, the very value, of empirical science? Practically every single thing that has made life measurably easier for the great mass of people over the last few hundred years has been developed by that process … observe, theorize, test, revise.

I am not claiming that no children learn this … but I am skeptical that even a respectable percentage of them do. Not anymore.

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