Col Not Colin

Hi everyone. Firstly, thanks to @wtf who has provided plenty of reading material. It'll take me a few days to browse through that lot. Admitting that I haven't read much of those links, what follows is just an initial reaction or speculation: What do you mean by a proposition about the naturals? Does it ONLY have to involve the naturals, or can the proposition just involve one natural number somewhere? For example: Take a typical undecideable statement (like something examined by Godel), let's call that proposition P, and just add an obviously provable statement like Q = "2+2 = 4". Then, using ^ as the logical connective for AND, the statement P ^ Q would be a new statement that is undecideable and involves natural numbers.

Hi bayukutten. I would have thought most maths textbooks for a school Mathematics course for people between about 11 and 15 years of age would have a section on signed numbers (often called positive and negative numbers). I haven't reviewed all of these books so I wouldn't like to single out one book as being better than others. You obviously have internet access. I find the BBC Bitesize educational resources are usually of a high standard and free to access. So I would recommend their site: This is aimed at British key stage 3 (age range 11 to 14): https://www.bbc.co.uk/bitesize/guides/z77xsbk/revision/1

Understood and apologies to the OP. A new thread is not required, enough info on that side-line has already been supplied. Thanks. Returning to the original post: It is interesting and highlights a natural human curosity about the nature of time. The replies already illustrate the difficulty in finding an explanation that seems satisfactory and frequently involve abandoning one's original view of what time is supposed to be. Hence, trying to understand what makes a human being aware of time passing seemed relevant for the purpose of finding a more satisfactory resolution BUT this is not required and there is no need to take the discussion down that road. ---- End of side-line discussion, waiting for further input and direction ----

thanks iNow. +1.

Condolences @MigL Hi Time Traveller, hope you are well. I think several people have already mentioned what time is assumed to be for most models in Science. Time is another co-ordinate we need (in addition to the usual spatial co-ordinates) to specify where events happen in spacetime. Any other property or nature of time is usualy less important than you would think, or else the model you are working with will tend to specify what time needs to be and what properties it is assumed to have. You (we) only need to have a passing knowledge of relativity to realise that a scientist's ideas of time have undergone a lot of changes in the years from Newton to Einstein. None-the-less, time does seem important, very special and uniquely different to spatial co-ordinates for human beings. There are scientists investigating the nature of time directly and many more in diverse fields that uncover a bit more information that seems to connect with our understanding of time. There are some models that utilise geometry only, without any explicit time parameter and then an emergent property is exhibited which seems to behave like time. These mesh fairly well with the ideas you mentioned in the OP. In particular, in those geometric models you don't really need a thing called time because there are many other parameters you can use to form derivatives or calculate rates of change with respect to instead. I have a suspicion this thread will echo many other discussions about time. To make a slight change, I've got an interest in what we (human beings) use as an internal clock. I wondered if we can get a Biologist to make a comment or post a link to explain the current best theory for how we are aware of time? I assume it's some chemical reaction occuring in some group of cells and that it may proceed at a different rate according to factors like temperature. Does time seem to pass at a different rate when you are hypothermic? @CharonY Do you know a biologist who might post a link for this? (or shall I take my chances with a search engine).

@studiot thanks again, don't do anymore, this stuff must take you ages. I'm getting to grips with the stuff and there's plenty of online resources already. I've still got to try out the software you mentioned the other day (but one thing in a day is enough for me, so that'll wait). @Phi for All A question / small request. The sandbox prevents editing old posts after a lunch break (? about 50 minutes ?). Worse than this, if you go and Copy an old post then the LaTeX doesn't re-appear when you paste that into a new comment box. So I can't use the sandbox as anything like an online assembly place for posts involving a lot of LaTeX where I may need to take a break and continue tomorrow. Is there an easy solution? If not, I guess I'll just try and keep a copy of the LaTeX somewhere else because it's a bit sad to have an hours work you can only look at and not do anything with. 😢

Latex test: [math] \frac {d^2 x^{\mu}} {d \alpha^2} + \Gamma_{\rho \sigma}^{\mu} \frac {dx^{\rho} } {d \alpha } \frac {dx^{\sigma}} {d \alpha} [/math] Same again, in larger font: [math] \frac {d^2 x^{\mu}} {d \alpha^2} + \Gamma_{\rho \sigma}^{\mu} \frac {dx^{\rho} } {d \alpha } \frac {dx^{\sigma}} {d \alpha} [/math] Test post: The following is the usual geodesic equation from General Relativity: [Eqn 1] [math] \frac {d^2 x^{\mu}} {d \tau^2} + \Gamma_{\rho \sigma}^{\mu} \frac {dx^{\rho} } {d \tau } \frac {dx^{\sigma}} {d \tau} = 0 [/math] Where [math] \tau [/math] is an affine parameter (for example, proper time) and the path is given by [math] x^\mu = x^\mu (\tau) [/math] Suppose the same path is parameterised another way. Let [math] x^\mu = x^\mu (\lambda) [/math] where the parameter [math] \lambda [/math] is NOT assumed to be an affine parameter. Then, [Eqn 2] [math] \frac {d x^\mu} {d \lambda} = \frac {d x^\mu} {d \tau} \frac {d \tau} {d \lambda} [/math] by the chain rule (we can assume [math] \frac {d \tau} {d \lambda} [/math] exists). Hence, [Eqn 3] [math] \frac {d^2 x^\mu} {d \lambda ^2} = \frac {d^2 x^\mu} {d \tau ^2} {\frac {d \tau} {d \lambda}}^2 + \frac {d x^\mu} {d \tau} \frac {d^2 \tau} {d \lambda ^2} [/math] Combining Eqn 1 and Eqn 2 we obtain, [math] \frac {d^2 x^{\mu}} {d \lambda^2} + \Gamma_{\rho \sigma}^{\mu} \frac {dx^{\rho} } {d \lambda } \frac {dx^{\sigma}} {d \lambda} \space = \space \frac {d x^{\mu}} {d \tau} \frac {d^2 \tau} {d \lambda ^2} + \large [ \normalsize {\frac {d \tau} {d \lambda}} \large ] \normalsize ^2 \large ( \normalsize \frac {d^2 x^{\mu}} {d \tau^2} + \Gamma_{\rho \sigma}^{\mu} \frac {dx^{\rho} } {d \tau } \frac {dx^{\sigma}} {d \tau} \large ) [/math] COMMENT: having a time-limit to edit posts in the sandbox seems un-kind. Sorry for making a lot of posts, I ran out of time. Crumbs... this is hard work. I'll try an equation editor and see if I can import the finished thing.

[math] \frac {d^2 x^{\mu}} {d \alpha^2} + \Gamma_{\rho \sigma}^{\mu} \frac {dx^{\rho} } {d \alpha } \frac {dx^{\sigma}} {d \alpha} [/math] Same again, in larger font: [math] \frac {d^2 x^{\mu}} {d \alpha^2} + \Gamma_{\rho \sigma}^{\mu} \frac {dx^{\rho} } {d \alpha } \frac {dx^{\sigma}} {d \alpha} [/math] Test post: The following is the usual geodesic equation from General Relativity: [math] \frac {d^2 x^{\mu}} {d \tau^2} + \Gamma_{\rho \sigma}^{\mu} \frac {dx^{\rho} } {d \tau } \frac {dx^{\sigma}} {d \tau} [/math] Where [math] \tau [/math] is an affine parameter (for example, proper time) and the path is given by [math] x^\mu = x^\mu (\tau) [/math] Suppose the same path is parameterised another way. Let [math] x^\mu = x^\mu (\lambda) [/math] where the parameter [math] \lambda [/math] is NOT assumed to be an affine parameter. Then, [math] \frac {d x^\mu} {d \lambda} = \frac {d x^\mu} {d \tau} . \frac {d \tau} {d \lambda} [/math] by the chain rule and since we can assume [math] \frac {d \tau} {d \lambda} [/math] exists without loss of generality.

Thanks @Phi for All and also @studiot Any hints for the fastest way to get a suitable delta ¶ ? Hmm... given up pasting the "symbol" font into here, looks like you convert to your own font automatically. What is that font? Charmap would be more useful if I pick from the same font you use, I think. Anyway, my intention is to create some partial derivatives and /delta in LateX will curve the wrong way. I'm guessing partials are created with /frac but if there's a faster way, like /partial I'd like to know. No rush to reply, I can go and read the manual on LaTex or similar links given by the OP. I'm also looking into the suggestion studiot made earlier (not on this thread) of just using another piece of software, which seems considerably easier. It's just I don't always have my desktop PC but the built-in LaTex of this forum should always be there wherever I am and whatever computer I'm using.

[math] \int _0^{\infty} x^2 dx[/math] [math] \frac{\delta^2 x} {\delta a \delta b} = frac{\delta^2 y} {\delta^2 x} [/math] Did I hear someone say there was a sandbox? How do I post to that and check my formula?

On a minor note, how are we all identifying "the least number of assumptions"? A Theory in Biology can use a small number of words and what appears to be a small number of assumptions. However, the moment any of that is something like "and this keys to the recptor site", there are actually a thousand-and-one small assumptions attached. There are assumptions that microscopic objects like molecules behave like macroscopic objects. That ping-pong ball models of molecules actually convey information about shape. Within each molecule the inter-atomic bonding is also represented by some generalised theory of bonding. In particular, it was never assumed necessary to find exact solutions to any Schrodinger's equation for the molecule. If Ockham's razor is applied as a judgement tool for science then Chemistry isn't far removed from wishful thinking and Biology is something akin to superstition. However, Biology is obviously an extremely valuable science. That is the one that has developed vaccines for the Covid-19 virus. It has demonstrability of predictions and utility for human beings "in spades" and that is what makes it a high value science. If we had been determined to make the least number of assumptions, then computers would still be trying to find numerical solutions to wave equations today and human beings would not yet have had the understanding that molecules in the Pfizzer or AstraZeneca vaccine could even hold together.

HI everyone. Obviously I quite like Ockhom's razor and a bit of reductionism but I'll play the part of the opposition for the moment. Thanks for the history, studiot, that was interesting. Has Ockham's Razor become blunt in the last 700 years ? Maybe it's not been made blunt enough. In theoretical physics there is a drive to develop unified theories and also to identify the most fundamental structures and assumptions required to create what we observe around us. Such a theory has value and may also be fine for a computer to understand and utilise - but it's unliekely to provide understanding to a typical human being. There is value in science that focuses on providing utility rather than prioritising making the least assumptions. Ockhom's razor made sense back in a time when science was frequently mixed with superstition but in this modern age it seems less relevant. Since it is still used as a judgement criteria for scientific theories it is having a detrimental effect on the motivation of new science. Ockhom's razor still has a place but, as indicated by CharonY, the demonstration and verification of predictions should now be a more important requirement in the judgement of a piece of science (and the inevitable motivation for science that will come from that).

Hi. Here are some answers: 1. The entire book was handed to me, not just that one section. However, that section is the most relevant part and copying the entire book to this post would probably violate copyright etc. I certainly didn't read all of the book at the time and probably only about half of it to this day. 2. No I didn't ask for this specific information. I was interested in set theory and real analysis. 3. I was not aware the book existed. 4. What actions prompted the other mathematician? This is speculation, I'm not the other mathematician. I suspect it's that I showed an interest in the fundamental nature of mathematics and some desire to make sense out of what we (Mathematicians) have been doing for the last few thousand years. 5. Practical context: Final year project in an undergraduate degree. The other mathematician was my supervisor. I don't seperate my learning environment from a work environment too much because that is the area I went into shortly afterward as my work environment (I became a lecturer). 6. I expected to learn about the minimal assumptions required to construct the Real numbers. I ended up learning about logic, set theory and the ability for Mathematics to analyse itself. 7. Yes, I have some interest in notation but in the broader context of identifying the fundamental nature of Mathematics. Another thing that made me reply was that you had few replies. 8. You can message me about NSA although I am NOT an expert in that area and these days I always have housework and real life stuff to do. I'm only here at the moment because the covid situation is essentially cutting out all other avenues for discussion (or anything). I would also advise caution about applying too much of your time to studying NSA. As indicated by wtf It is widely regarded as an inefficient method for real analysis. However, it's a great thing for pure mathematicians to have some familiarity with. I'll try and figure out how the message system works and create a short "test".

Hi everyone. A new YT science video creator is ScienceClic. Original videos are in French but they are re-releasing most of them in English. They tend to specialise in Relativity, especially General Relativity. Positive Comments: It's new, rapid, conveys some good ideas, uses some new animation techniques and it isn't afraid to change some well established concepts if it helps understanding. Actually, changing these concepts is a bold step that may be a way forward in the pedagogy. Neutral Comments: It doesn't follow historical development but instead seeks to present the finished article that is modern GR efficiently. Negative Comments: It isn't afraid to change some well established concepts and makes no mention of this. There is a noteable lack of rigour constructing Christoffel symbols and the geodesic equations with some thoroughly shameful exploitations that could leave you screaming. The pedagogy for GR may be improved but at what cost? Overall: 8 / 10.

Please check the items written in red There are 1x10-3 micrograms in 1 nanogram, if that's what you meant. On the other hand, there is some information missing. You started with 4.5 nanograms which is just the mass of something. You are after a final answer that is a concentration (mass per volume). It seems likely that you were told something like "there are 4.5 micrograms per VOLUME OF MEASURENT (maybe 1 litre)" and then asked to convert. IF everything is taking place within a volume of 1 microlitre, then what you have done so far is fine. However, if the original information implies the 4.5 nanograms were in a different volume, then you were absolutely correct in thinking there would be a second step - we must also adjust for the change in volume units. Let's give an example only: If you started with 4.5 ng in 10 litres, then that is equivalent to 4.5 x 10-4 ng per micro-litre, which is 4.5 x 10 -7 micrograms per microlitre. Minor note, there is also a section called "homework help". People usually respond quite quickly and give guidance that is more helpful to you if you are learning.

@iNow ty and +1.

Hi everyone. Request for advice: This is a book talk type thread. Is there a similar thread for YouTube videos? I'd like to see someone talking. I've been stuck in with a covid lockdown too long and I want to see a person talking some science. (I also have a burning desire to complain about one youtube video I recently watched - but that's another issue).

Hi. I hope you are well. Is that a typing error or did you really mean it is NOT like a fabric? This also has some connections with a thread started by Mordred in the General Relativity section. Specifically space is NOT filled with a fabric material. Here's a link to that discussion: Your last question was "Have you understood why gravity exists?" No, not really. Few people like the stretchy fabric model as demonstrated in that video. It's full of problems. Here's the two obvious ones: 1. There isn't any "fabric" in space. 2. The distortions were created by gravity and not really directly by the mass. If you took that stretchy fabric into outer space well away from any gravitating source and placed some mass on it, it wouldn't bend and distort. People argue you are using gravity to try and explain explain. However, looking on the positive side. That stretchy fabric analogy has been used for years and it' OK. It does get some ideas across to the audience.

Thanks iNow and Swansont.

OK. 1. Rest assured, I wasn't telling the OP that this is the best way to learn calculus. Although... at some later time, I feel I could make that argument but I've got dishes to wash first. 2. I'm quite delighted that someone else knows anything about NSA and I'm most impressed with your post. 3. Yes the axiom of choice is an issue BUT it's a a set theory issue and that's great because set theory was also relevant. You need a set theory just as much as you need a Language. 4. I'm not sure you require the Ultrapower construction (or an Ultrafilter) just to construct an object that is like the hyperreals, especially if we soften the requirements on * slightly. To be more specific, appendix 1E in my old copy of Keisler entitled "a simple construction of the Hyperreal Numbers" seems to construct a triple (R, R*, *) such that these things A~D will hold: (A) R is a complete ordered field. (B) R* is a proper ordered field extension of R. (C) * is a map from { functions of n variables on R } to { functions of n variables on R* } . * : f --> f* and, the field operations on R* are the image under * of the field operations on R. (D) If two systems of formulae have the same real systems, then they have the same hyperreal solutions. (mapping formulae and field operations by * as expected, but it will make the definition too confusing if I shove it all in here). That's a thing, a simplified thing, we could call the hyperreal numbers and I'm fairly sure that ultrafilters were not required. You did still require the Axiom of Choice (or Zorn's lema). The * transformation won't take all first order sentences across to R* but it's not too far off. 5. The Hyperreals are non-Archimedian. Yes, I agree but ... who said it was and does it matter? It's interesting but not a problem. Thanks for that and I will have a look through. WHAT IS THIS THREAD ABOUT? Well, the OP is gathering information from Mathematicians. They were interested in things like how Mathematicians communicate. Getting some concrete examples of this, like the post you have just sent as a reply to me etc. You're probably in-line for inclusion in the study, so be warned but don't worry too much. Any post can be viewed by anyone and data could be gathered, we have the advantage of knowing it is happening. What are the OP's goals? Various and you should probably read their posts instead of getting a corrupted opinion from me. I would say they have an interest in designing software and determining if notation is important. Anyway, +1 for your post. Anyone who knows a little NSA can't be all that bad and I'm very grateful for the discussion.

Hi everyone and presumably Hi from everyone else. I'm also fairly new here but I'm not sure we can have a whole forum thread full of "Hi". I'm not sure how this voting or rating system works either but Yance can have a +1 from me for being brave enough to use the forum. BigR can have +1 for being brave enough to reply. No more +1 from me, regardless of who joins in the discussion, since I don't think we're suppose to hand them out like smarties - but how would I know? Request to Moderators, staff and anyone else: Can we have some guidance on what Newbies can, should or may like to do, please? Also could you make it BIG AND BRIGHT on the screen just after we create an account, so that a simple person like myself can find it. Additionally, could you just automate a +1 for each newbie after they've either created a post or replied on an existing thread without it needing to be taken straight off the thread by the moderators. (Because that is a fair achievement). OK, I know I'm asking a lot. Meanwhile, may I suggest you (Yance and BigR) head over to the Forum Announcements section? It doesn't actually help much but it's there and it seems like a place you should go for guidance. You can imagine that you are passing other virtual newcomers to the forum - those returning may still have the same puzzled faces they went in with but that's beside the point. More usefully, may I ask what you want or hope to find? Maybe I've seen it, or someone else will reply. From what I've seen so far, this thing is reasonably open and free-form. Try and get your own posts in roughly the right slots (for example Chemistry and Physics have different sections and there's also a SPECULATIONS section for wild speculations about anything), or else just browse through and join an existing discussion. @swansont Not sure if this is how you message a moderator and I'm sorry, Swansont, for putting your name here. You were one of the people listed in the forum announcement section. Please could I ask for a link to some information or some brief guidance for new-comers to this forum? Good Luck to both of you and best wishes.

Hi. Well, as promised, here's (a bit of) something that was passed to me by another Mathematician. What is it? An extract from a book, it was the actual physical book that was passed to me. Formal reference: H. Gerome Keisler, Foundations of infinitessimal calculus, Publisher: Prindle, Weber & Schmidt, incorporated, 1976. An online version exists, although there's obviously been a serious re-juggling of pages and content since I had my hard copy (we called it a book). The images in this forum post are screenshots from the appendix of that online copy. You'd probably prefer to point your data collection tools at the electronic copy and these images are then pages 178~180 as described in that text but pages 188~190 as described by my pdf reader (since some introduction and cover pages are counted). Here's the URL at the time of writing: https://www.math.wisc.edu/~keisler/foundations.pdf What does it mean? Well, it's a theorem and proof showing how the hyperreal numbers relate to the ordinary Real numbers. That's important for an area of Mathematics called Non-Standard Anaylysis but it's not what made this important to me. Why was it important to me? It's just a background story. A bit like "the little boy who cried wolf" is a background story that seems to be an awfully wasteful way of explaining to children that they shouldn't tell lies. What was important about this section of the book was that formal first order logic and set theory was used throughout the construction of the hyperreal numbers. It's when I started to appreciate that this might be all we need for mathematics. For example, sets might be the most fundamental structure in mathematics - a bit like atoms for the Physicicists. The actual symbols and notation used in mathematics are not important but that there is a language and some structure to that language may be the only important thing. Definition: A Language is a set of symbols appropriate for the structure under consideration. A first order Language must possess the following symbols: Connectives, Variables, Commas, Paranthesis. Connectives must include the following symbols "Implies" ; "if and only if" ; "Not" ; "And" ; "Or (I'm skipping the rest of a definition for a Language, you can find a full description in the book by Keisler if you're interested. What we have above is enough to get a flavour of the idea). Typically we draw a little arrow -> for the implies sign and a double-headed arrow <-> for "if and only if" etc. etc. BUT the exact choice of a picture for that symbol is entirely up to you, their Boolean logic values are all that matters and that is what makes it an "Implies" symbol (and they are, of course, exactly what you would expect as a computer scientist). With this idea of a language, what the Elementary Extension Principle is saying is quite simple: A sentence of first order logic that is true in the Reals will translate to a sentence that is true in the Hyperreal Numbers. I don't have any post-it notes or diagrams scribbled on the back of an envelope from that Mathematician but I don't thnk they would have helped much anyway. I think I needed to read the full fairy-tale of Non-standard analysis to learn about Logic and Language and re-evaluate what this thing called "Mathematics" might actually be. STOP here. You said one thing per post is preferrable. I've probably already covered several different things by mistake, sorry. DATA, attachments etc. appear below: Image, part 1 of 3. Image, part 2 of 3: Image, part 3 of 3:

ML = Machine Learning ? Does this research have any connection with Microsoft's Lean? Is that the ML abbreviation? If not, you may have a passing interest anyway. I'm cautious about giving out links but you sound competent enough to make your own searches for what is being done at Imperial College, London, UK - where Lean is being used to try and develop a competent artificial mathematician. For whatever it is worth, I was very relunctant to click the links you provided, especially the hyperlinked word this you used in one earlier post. I prefer to be in direct control of navigation and not pass extra information through the URL string sent to a server - but then I am old and don't own a mobile phone. I wouldn't really want to comment or offend anyone or any organisation. On the other hand, listing some positive points seems less problematic: Studiot, the other person replying to you seemed very pleasant and I probably wouldn't have bothered coming back to this forum for a second day if there hadn't been a few like that on here. Physics Stack Exchange is also another good site. I promised you an example, I'm sorry, Real Life has got in the way and it took me half-an-hour to write this much text. Another day please.

@studiot Thank you very much.

Well, I guess it means the OP found an answer from elsewhere faster than you could type. There is a piece of advice you can give me though. How can you get those formulae in this textbox (even in the same day)? Are you dragging and dropping them in from some other software? What can a simple man like myself use for a bit of mathematical notation? That empty quote is what turns up when I highlight just the last part KCl(aq). What is this stuff? [Late editing: Wow, it appears when you post it]