studiot

Great summary +1. @Trần Thành 2022 I can only add that I suggest you look up 'mass defect and chemical reactions'. Here is a good reference. https://chem.libretexts.org/Courses/Grand_Rapids_Community_College/CHM_120_-_Survey_of_General_Chemistry/2%3A_Atomic_Structure/2.07_Mass_Defect_-_The_Source_of_Nuclear_Energy Note this also explains swansont's good point further.

Good question +1

Yes, I think he did say that, or something very like it. Did you read his delightful book on symmetry containing references to his visit to the Alhambra ? The interesting thing about patterns is that they can be algebraicised for incidence using incidence matrices. Another related point is that in studying patterns maths uses much the same techniques as other disciplines. In particular idealisations are extracted from observations of reality, many of these idealisations are also of great use in Physics. Another author much concerned with pattern is Philip Ball - The self Made Tapestry. Di you know the pattern to determine the optimum level to fill a cement mixer ?

And yet, as I noted, it is the first statement to appear in Euclid! You can't have Pythagoras, without algebra. I don't see how much Geometry you can do without numbers. There is no similarity theorem in curvilinear space eg in spherical triangles. You can't have vector geometry without the zero vector as it is involved in four of the vector axioms. Therefore the link between extrinsic and intrinsic geometry (holonomy) fails as you can't have vector fields without vectors. Without a side of zero length, Aubel's theorem in plane geometry is incomplete. What I am saying is that 'zero' is buried very deep in Geometry so that we don't normally think about it, On the other hand I fully support this comment. There is plenty of archeological evidence that maths was in at the beginning of language. (see John Derbyshire "Unknown Quantity"). However perhaps 'counting' is too sophisticated a word and a deeper, but incredibly useful, mathematical technique of tallying (putting in one-to-one correspondence) came first. Perhaps this process even came before real words separated out from grunts which simply meant "Attention!" I don't disagree with this at all. Maths is a tool. And we'd rather use maths to make a hammer than use hammers to do maths. I even think maths is at the basis of language. Even people who say they hate maths, I think, have a simpler, more basic way of mathematically understanding the world. Perhaps less sophisticated, refined, or whatever. May I respectfully remind both you and @MigL that this thread was started by a professor of Mathematics, in the Mathematics section ? Whilst I would wholeheartedly agree that maths is an incredibly useful tool in many disciplines, especially Physics, this is surely a question of Mathematics?

But that is part of the definition of a distance function (the Mathematical term for a metric) Without zero you have no metric. But that is not the only use of zero in Mathematics. You probably know the four colour theorem, and the two colour theorem. Can you draw a map with zero colours ? Or perhaps you would like this poem I have highlighted the use of zero bend (curvature or the reciprocal of radius) to indicate a straight line. You also need zero in projective geometry for the ratio theorem to indicate the 'missing' ratio.

I do understand you and please be clear. That is just not possible. There are many branches of maths. You need a bit of almost each and evry one to study fully any one branch. That is why Maths in particular is taught in what I call a spiral approach. At each turn of the spiral (say each year) you learn some more of some or all the branches, based on all that you learned in each branch the precious year. Mathematicians carry this on through their working lives. Here is a true story. When I was in junior high I was pretty good at Geometry. I thought I knew everthing there was to know about Geometry and could solve any problem. So when I move up to senior high (college for you like the american system) I was suprised to find the first thing on the curiculum was - Geometry. But Coordinate Geometry. I was shocked and blown away by all this new stuff. But it underlies calculus, applied maths, differential equations, physics and so much more.

Hello, Riba and welcome. Maths on your own huh ? That's both ambitious and admirable. So is asking questions so remember to come here and ask if you get stuck. Quadratic equations and basic trig ? So you are in junior high then. You haven't said which country you are in, but in the UK you will be doing some algebra (those pesky equations), some geometry, which goes with the trig, some modern maths such as sets and logic. There will also be lots of practice applications questions (both in maths symbols and in words). This is all good stuff that leads into later material, normally encountered in senior high. This is where you will encounter calculus first and perhaps a taste of differential equations. But you will also encounter coordinate geometry and other topics. There is no one order to 'do' the different topics in. This is because Maths hangs together so you need a bit of one branch to be able to work in another branch. With that under you belt you can use your new knowledge to expand both branches and perhaps a third, fourth etc. One word of warning since your ambition is (astro)physics. Any course whether at school, college or online will be designed for the questions asked to be solvable using the material taught. But no course teaches it all so they leave out the exceptions and difficult pieces, and never ask questions about them. So it is easy for a student to be lulled into a sense of false security, thinking they know enough. But in real Physics it is often the exception that you need. Go well in your studies, I look forward to periodic questions from you.

Euclid Definition#1 "That which hath no part." This is an early version of a metric which you cannot have without a zero.

Less true than it used to be. When I was first at university many of us brewed beer in plastic buckets/small dustbins. Yellow was a very popular colour. Then some professor (at Southampton I think) pointed out that the yellow colour is given by cadmium and this dissolves out into the beer. Cadmium, of course is poisonous.

Well that would make all mathematicians plagiarists. But then I suppose we are very fond of plagiarizing the phrase 'if and only if'

Can't see why. Swansont has already told you the essence of how to do it, collect enough energy from the wind. As an observation he also said So a 'lighter than air' balloon would make the collection burden easier.

Personally I'd be more worried about the chemicals allowed in bottled 'beverages' than either salt, soap or vinegar. More especially if you drink a lot of them.

Lots of great comments here. +1 I will add to just two of them. 1) Dry suits can come with weighted boots, especially those meant for fixed air lines. 2) I had to drop my weight belt once in an emergency. It was an expensive exercise. However if you are multiple diving diving a fixed site (perhaps with a line) you can leave your weightbelt on a line or for others to haul in or even have a system of hauling up bottom sourced weights with each dive, and dropping them back again at the surface.

Is this a genuine puzzle/brainteaser or is this a genuine physics question ?

What aspect of GR makes it a field theory ? In other words what properties specified and used in GR form this 'gravitational field' ? It should be noted that when GR was published in 1915 it described no such properties and did not conform to the then still developing definition of a Field Theory formalised by Kellogg in his 1929 book "Potential Theory". This definition was in use for most of the 20th century. In order to turn GR into a Field Theory it is necessary to modify both GR (via a non linear Lagrangian) and the definition of a field (theory). Quite a lot of progress has been made towards this in the last few decades and there are postgrad lectures on the subject at MIT, including youtube lectures and padf papers. https://www.google.co.uk/search?q=Donoghue+effective+field+theory&source=hp&ei=SGfdYZDOCbyEhbIP06GW8AI&iflsig=ALs-wAMAAAAAYd11WKFROtIycqoYvDENHbw-KziO8FkI&ved=0ahUKEwiQktPJyan1AhU8QkEAHdOQBS4Q4dUDCAg&uact=5&oq=Donoghue+effective+field+theory&gs_lcp=Cgdnd3Mtd2l6EAMyCAgAEIAEELEDMgUIABCABDIFCAAQgAQyBQgAEIAEMgUIABCABDILCC4QgAQQxwEQrwEyBQgAEIAEMgUIABCABDIFCAAQgAQyBQgAEIAEOgsIABCABBCxAxCDAToOCC4QgAQQsQMQxwEQowI6CAguELEDEIMBOg4ILhCABBCxAxDHARDRAzoLCC4QgAQQxwEQowI6CwguEIAEEMcBENEDOggILhCABBCxAzoICAAQsQMQgwE6CwguEIAEELEDEIMBOggIABCABBDJAzoFCAAQkgM6DgguELEDEIMBEMcBENEDOgUIABCxA1AAWJ5jYP5maABwAHgAgAGmEogBuDSSAQkwLjQuOC0yLjGYAQCgAQE&sclient=gws-wiz A new term "Effective Field Theory" has been coined, but note the caveat (from Wikipdia) it is an approximation to the real thing. @geordief you have +1 for one post, have +1 for the other as well for being brave enough to ask rational questions. Edit I should add that Effective Field Theory is not quite there yet, nor is it (yet) mainstream. But it does answer Markus' points about scale, local v global variables and quantum considerations.

So just as I said Lecture 1 contains many things, including analysis and synthesis.

Yes, and when I do (it's nearly one hour long) I will not misrepresent what he (or anyone else) says, and I would expect to learn something I did not already know, even though it was only lecture 1 of 30. I have already said that it seems a good course, and its also free. I note from the transcript that both analysis and synthesis are defined and contrasted.

Studiot Genady This most definitely appears to contradict my comment yet Stanford I have underlined the relevant words in the topics list appearing under lecture 1.

I did indeed state that fourier transforms are applied maths. You have contradicted that statement more than once now. I have also pointed out that the very words in the link you provided state explicitly that fourier trnasforms are applied maths. I even quote the passage from that link. Yet you seem to maintain that fourier transforms are not applied maths. Have you ever studied fourier transforms at all ?

Then you were not clear enough.

So you disagree with the quoted statement from your own link ? It refers to manynthings. Please don't sidestep the issue by implying that there it refers to only one matter. Did you ever take this course ?

In an affort to stay on the topic of emergence, I looked carfully at the Wikipedia entry on emergence. https://en.wikipedia.org/wiki/Emergence It looks like other members have also looked there since I note several terms (words and phrases) that have been mentioned here. Two things stands out. A) Apart from also discussing emergence in non scientific contexts, Wiki offers several ancient and also very modern contradictory accounts and definitions of emergence in the the Sciences. B) About the only characteristic of these different emergent characteristics seems to be the notion of emergence being the result of a combination/interaction of parts which do not themselves possess this characteristic. This second point begs the immediate questions How many parts are required? Is the process reversible? That is can the emergence disappear if we revert to a collection of the parts? Can we actually revert to or recover the original parts? However though Wiki is rich in general statements, it is poor in detailed examples. So taking my exples of an arch and an atom bomb and comparing them I note For the arch the answers are Minimum 1, no upper limit Yes the parts may be deconstructed and reconstructed as many times as required Yes For the atom bomb Many parts are required. The actually number is statistically detemined. No No The actual parts are destroyed in the emergence

I don't think anyone suggested they are applied maths. Fair enough, not me then. Now how about properly addressing my points and making a discussion of it ?

Well I think you implied that in your words, so I highlighted the word 'pure' - which you did use - in my reply. Pure Maths, by definition, does not include Applied Maths, which you also referred to in relation to 'concepts'. I don't think I defined anything so I am not sure about your last line but taking your second line I would say that vector spaces are the "pure maths", and fourier transforms are the "skillfully operating with it to get deep and rich theorems" In other words the applied maths. Thank you for this reference, I expect it is an excellent course. Edit A quick look at this course suggests that it is pretty comprehensive. It also states in so many words that this is an applied maths subject, firmly based in StanfordEngineering Everywhere no less. Lecture 1 refers to 'analysis' and 'synthesis', a subject distinction you to seem to wish to avoid. When I studied the Integral transforms in general I was following a part of my math degree called 'Linear Analysis' which had both pure and applied components. The pure componet was represented in the course texts by Nering and by Hoffman and Kunze. The applied component was represented by the text by Keider, Kuller, Ostberg and Perkins. Needless to say the integral transforms came in the applied component. However my main objection to your 'rule based' approach is that it is 'static'. Adopting this therefore precludes the study of process in Mathematics, process being a dynamic object. This approach is therefore akin to studying only statics in Mechanics, and ignoring dynamics and all that dynamics introduces. Finally another member recently tried to categories the parts of Mathematics, you may wish to look at their thread.

Yes, I do. So why did you list the Fourier transformation as a fundamental concept ? It is simply a skillful operation of the vector space concept.