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If you just copy, you don't understand.

A quick "google" with give you several definitions of mathematics. To me they tend to be necessary, but far from sufficent. As example; Mathematics is the abstract study of numbers, shape, structure and change. Numbers are part of mathematics, so are simple geometric shape, by structures we mean patterens and relations between them, and differential calculus ("rates of change") are all parts of mathematics. This definition seems okay, but a little vague and it may not be very clear what we mean by a structure. Also, as mathematics evolves, solutions to problems often come from outside the area the problem was initially defined, any defintion must be able to include any future mathematics. It maybe better not to define mathematics by what it studies by how it studies them. A first attempt at this would be to reduce all mathematics to logic, but this fails. Mathematics as a science is also a tricky issue. There are many parallels between how a mathematician works and the philosophy of the scientififc method. Anyway... My question to you is what is mathematics?

Symphonically synchronistic ✌️

In the case of plagiarism, they're not taking the lecture, or copying what the teacher wrote on the blackboard. They're given an assignment to research a topic and produce their own report on it. Instead, they are copying another person's report, without doing the research, which was the path to understanding. Therefore, understanding does not enter into the issues surrounding plagiarism. That's completely beside the point. It's not about great pedagogy or understanding. They want the paper credits: the certificate, the diploma, the official recognition that they have completed a course of studies, so that they qualify for a position. But they have not actually learned the material, so they will not be competent in that position. You'd better hope none of them are your pharmacist!

Amazing. I was listening to this song no more than 3 or 4 days ago. I've just remembered and was about to post it. And I see your post. !!

Can a baby step be taken?Can it be shown that the smallest imaginable system can arise out of some more fundamental process? In a deterministic ,reproducible way... (Not sure what I am asking,but we agree surely that the links between an individual quark and the movement of the left ear of a giraffe are impossible to show in practice and perhaps in principle) I think the idea I think someone may have brought up.of replacing spacetime distances with other forms of correlation sounds very interesting (exciting?)

On the other hand (and more recently) NASA also says, contradicting https://www.nasa.gov/feature/goddard/nasa-study-mass-gains-of-antarctic-ice-sheet-greater-than-losses - from Grace Satellite gravimetric data - And - I am not sure how the different data is reconciled. I would note that data based on ice and snow surface elevations have innate potential to be misleading about mass changes - I'm inclined to think the gravimetric data has less room to mislead.

As to emergence there is, I think, a dilemma between principle and practice that I think overrides almost any other consideration. Directionality of emergence is very clear in principle, but there are insurmountable difficulties in practice to ellucidate causation. Atoms make a dog. Dogs don't make an atom. It's what Weinberg called 'arrows of explanation.' That's very clear in principle. Even though it's impossible in practice to tell anything about dogs from the laws of atomic motion. It's very clear to me that dogs emerge from atoms; atoms do not emerge from dogs. I happen to know that some very philosophically-minded people think otherwise, which are the ones that @TheVat characterises as 'strong emergentists.' I think they can do that, only because the 'arrows of explanation' are invisible to all intents and purposes. It's a hopeless problem, so there is room for people to exploit this practical disconnect, interpret it as fundamental, and what's more, invert the 'arrows of explanation.' In this example, I think people who hold this view are disregarding an approach that's much more plausible: feedback mechanisms. Those are compatible with molecular determinism, IMO. Even though they're extremely complex. An algorithm to run on a machine that proved beyond any doubt that there must be such a thing as a dog based on the quantum laws of motion. That would be a sight to behold. But I wouldn't wanna be the person analising the data. This would-be machine would have to prove the logical necessity (from the atomic laws, to be kept in mind) of giraffes, and T-rex, and gut bacteria, and... covering all the organisms that ever were, that ever will be, and that would have been.

Well, kind of. But I haven’t, in my own mind, arrived at a rigorous definition of the concept just yet. I don’t know the answer to this - personally I prefer to think of the lower levels as kind of a boundary condition to the ‘higher’ laws. Consider, for example, the laws of evolution - clearly, they are closely connected to lower levels, but are they determined by them? Can you start off with - say - statistical mechanics, and eventually arrive at the laws of evolution, perhaps through a simulation? Or how about the laws of psychology, sociology, or macroeconomics? Are they derivable from, say, the Standard Model? I think these are important questions to ponder.

Ring the bells that still can ring Forget your perfect offering There is a crack in everything That’s how the light gets in

I think one should not see philosophy too much as a separate subject, but looking in a special way to a subject. When a physicist is trying to find a particle at CERN he is doing physics. When a physicist is trying to find a new theory he is doing physics. Both activities are about physical reality. However, when it e.g. turns out that a conceptual framework does not work anymore (e.g. rise of quantum theory in the 1920s), when there are questions about the validity of certain methods, or about a demarcation criterion for science (e.g. string theory, multiverse) then one is doing philosophy. And one does not necessarily need a philosophical education for that: the interest in conceptual clarity and the intellectual capacity to do so, are enough. Latter should not be a real problem for physicists. First of course is really a question of what one is interested in. It's not everybody's thing. So not philosophers should push scientists to philosophical questions, so to speak from another discipline; the need for doing philosophy should arise in themselves because e.g. methodological or conceptual problems. Philosophers might be helpful in methodological and conceptual discussion, they are well trained in such discussions.