studiot

Since several people liked my comment, (Thank you all) , I will expand a little. The issue here reaches many parts of theoretical Mathematics, but one part goes to the heart of applications in Science. This is in probability theory and therefore in statistics and quantum mechanics. The probability of an event P(E) is defined as the limit of the relative frequency of that event as the number of trials tends to infinity. For instance consider rolling an n sided die. As the number of sides increases the number of different possible outcomes increases. As the number of possible outcomes increases so the probability of any given outcome (ie an event) decreases. So as n tends to infinity P(E) tends to zero. So we have the apparent paradox to resolve of how can we have a probability when we know that the die must end up showing one face or another, yet the probability of showing any one face is zero. In QM we resolve this by taking the probability between x and (x + δx) and taking a limit as δx tends to zero. In 'shut up and calculate ' mode we don't think about this we just do it and get 'the right answer'

This Zeno paradox is deeper than any of the others and was not properly answered for 150+ years after the others. The other Zeno paradoxes rely on sequences of integers and their reciprocals. This one relies on something deeper. The solution came after it became necessary to integrate many functions that could not be integrated by the Riemann integral, commonly taught in high school today. As you likely know, the Riemann integral is the sum of lots of small rectangles that make up the area under a curve. In fact it is the limit as the width of these rectangles ten to zero. But Zeno's question is what happens when that limit is reached ie the width is zero? The generalisation the the Riemann integral was introduced by Lebesgue (1875 - 1941) adn this ushered in what today is known as measure theory. https://en.wikipedia.org/wiki/Henri_Lebesgue The other approach to this issue was also developed in the first half of the 29th century by Paul Dirac and is known as the Dirac Delta function.

No, an engine will not run without it. An assembly as described without friction or any load will need at least an initial input of work (energy) to start. Thereafter it will continue in its state of motion as you describe.

I don't quite agree. Things are slightly more complicated than this. An ideal flywheel, crank and piston assembly by itself is a closed system. Yes. So it will continue its state of motion or rest indefinitely. But if you want to supply shaft work you require to input that energy somehow. And since some of the shaft work output of an IC engine goes to run absolutely necessary auxiliary devices, continuous energy input is required.

Another well balanced post. +1

You are mixing up thermodynamics and mechanics. No you do not need to put energy into a mass to start it moving. Stand under my window where the flower pots are and let me push one off the ledge. When it hits your head tell me how much energy I put into it. No the engine is not a closed system. Mass in the form of air/fuel mixture enters and echaust exists. It is known as a constant flow system or pseudo-closed since the same amount of mass exits as enters. But that entering mass brings (chemical) energy with it. So it is not an isolated system.The rising piston does work compressing the gas. The expanding gas then does work on the piston. Although work and energy have the same units and are different aspects of the same phenomen, there are subtle differences I suggest you look up. https://www.google.co.uk/search?q=difference+between+work+and+energy&source=hp&ei=3YPVYdboC43KgQbT-6qYCw&iflsig=ALs-wAMAAAAAYdWR7R2QBD6hUnxptAl9631h-LuTuAZn&ved=0ahUKEwiWy9faw5r1AhUNZcAKHdO9CrMQ4dUDCAg&uact=5&oq=difference+between+work+and+energy&gs_lcp=Cgdnd3Mtd2l6EAMyBQgAEIAEMgUIABCABDIFCAAQgAQyBQgAEIAEMgUIABCABDIFCAAQgAQyBQgAEIAEMgUIABCABDIFCAAQgAQyBQgAEIAEOgsIABCABBCxAxCDAToOCC4QgAQQsQMQxwEQowI6CAgAEIAEELEDOg4ILhCABBCxAxDHARDRAzoFCC4QgAQ6CAguELEDEIMBOgUIABCxAzoICC4QgAQQsQNQAFj8LmCSMmgAcAB4AYAB2ASIAdI6kgELNi45LjguMy41LjGYAQCgAQE&sclient=gws-wiz But +1 for accepting that you were not exactly correct before.

Yes that is true. So 1) There are numbers of no special consequence in Physics. That is there are parts of Mathematics that have no special meaning in Physics. 2) There are facts (numbers) in Physics which have special meaning that have no special meaning in Mathematics. Both leading to the conclusion that there is incomplete overlap between Mathematics and Physics. Is that not rational thinking ? OK I consider myself driven off the forum.

So What ?

What behaviour ? I said there are mathematically definable consequences. Would these consequences not be different if the value was ten times different ? Put into what ? Does the 'standard model ' predict the values of such numbers ? If so why bother to have them ? Why not just use the numbers themselves ?

+1 Where do you think this supposedly lost energy goes when the piston crown stops and then changes direction. Hint in order to stop it there must be a force and therefore there must be an equal and opposite reaction force on something else.

It is often forgotten that Physics abounds with data for which there is no theoretical basis or Law which states such and such 'must' have this or that value. A large amount of Professor Millikan's highly successful and readable book discusses the decades of impediment cause by a lack of knowledge of the value of e/m. Although we have now measured it and moved on, to this day we still can't demonstrate why it has this value and no other. Of course there is plenty of theory as to the consequences of this value. In fact pretty well every scientific equation and formula in existence contains such values ( mostly constants) which just are what they are and we have to measure them empirically. Physics, of course, is not the only Science where rational thinking holds sway. The description of crystal forms in granite given in Professor Swinnerton's delightful book is a masterpiece of rational thinking, without any Mathematics whatsoever in evidence.

It is worth noting that there are four pairs of North-South poles involved. 1) The N-S poles used for or coordinate systems eg Latitude and Longitude. 2) The N-S poles at the surface ends of the mechanical spin axis. 3) The geomagnetic N and S poles at the surface surface ends of the axis of an equivalent bar magnet, centred at the Earth's centre. 4) The magnetic N and S poles at the surface where the field lines are vertical. https://en.wikipedia.org/wiki/Geomagnetic_pole For most geological purposes, which is N and which is S make no difference. But for some including the weather and the aurora there will be differences. Life in general and particularly creatures however, will experience greater effects. Many creatures use some form of magnetic navigationand I believe even some plant life has magnetic orientation. Humans may experience some unexpected disruptions in addition. For instance the ground of electric power grids will be partially disrupted and may result in blackouts.

Thank you. That actually was a reference to a TV programme I saw about industrialised methods of growing tomatoes (and other veg), where they actually wish to exclude all creatures, other than human. So it was a reference to a Vegan world without any other creatures whatsoever.

A discussion with rancour. +1 If you are asking a question then clearly it is in your own interests to provide extra detail to those prepared to answer but needing to know more. Often their gift in knowing the subject better is knowing what questions to ask. If you are presenting a report on something eg in the scientific news section then the onus is on you to add sufficient summary to allow others to evaluate the subject presented. If you are presenting a hypothesis or conjecture, it is up to you to introduce such supporting material as may be needed, including answering questions or objections from the membership on the presented material. Material to introduce general discussion can be presented as a question or statement, either way, supporting background and explanation aids the discussion. Taking (or agreeing to take) one point at a time can be very productive.

Yes welcome, you have shown yourself to be a cogent thinker in whatever is your discipline. +1

+1 for a detailed explanation. I would like to add the following. Momentum will be accompanied by moment of momentum and momentum changes by (unwanted) torques. That is why, in the many arrangements that have been used including the one you proposed, the pistons are aranged to reciprocate about the driveshaft axis. Auto engines configurations and mountings have included the upright inline and transverse, the angled (between horizontal and vertical) inline and transverse, and the horizontally mounted 'flat' engines. Transverse mounting adds the requirements of changing the drive direction. Older aero piston engines went the whole hog with radially mounted multicylinder engines.

Roger Penrose is the son in another of those dynasties of scientists (including mathematicians). He is also foremost an analyst. I see litle to disagree with in your extract, though elsewhere in that book he lapses into his own unproved speculations, particularly about QM. His writing is, however, very dense, so one should always take careful note of the caveats he adds. I do however offer a counterexample to this unlimited statement. I have emboldened some key phrases. The only proof of the four colour theorem we have is basically the method of exhaustion. This method follows a different pathway from the one defined in the extracted passage. If you do not understand this please ask. I do not understand how you both agree and disagree with what I said ? You are new here so I will forgive you for this comment. I am entitled to make what civil comment I choose about your posted material and I will continue to point out where you are in error or misunderstand something. I will also explain in as much detail as you like why I think this to be so. That is what a discussion forum is for.

You clearly understand different structures of Mathematics and of language from the ones I understand. Without suitable structures and language little Mathematics and even less applications can be done. For instance those using applications tend to use the fundamentals of set theory such as uniqueness and closure without realising they do so. Group theory is based on such requirements They are not concepts they are requirements. The structure is organised this way for good reasons. There is nothing miraculous, fortuitous or gifted about this. It is quite deliberate on the part of Mathematicians. And it is from pure maths, not applied maths.

Thank you , now we are getting somewhere. This thread is discusses the language of Mathematics. Yet we are onto two pages and you have yet to mention set theory, mathematical processes, mathematical proofs ....... In answer to your question, analysis and synthesis are processes. Much of Mathematics at a fundamantal level, (and therfore its language) is about processes. When you analyse something you are working on something that is already there. You can describe it, categorise it, in many mathematical ways. You can put numbers to it, you can measure it. When you synthesise something you are trying to make or establish something that does not yet exist. In my experience that is generally a more difficult task. Mathematical processes such as proofs often fall into one of these two categories.

None of the above are any more fundamental than number theory is. I note from your posts that English is not your first language, although your English is very good. Is there some trouble understanding my posts since you are not directing your answers at the questions or comments I make ? If so I am very happy to expand on or clarify my comments.

Before you try to explain 'the galactic redshift', it would be wise to demonstrate that you know what it is you are trying to explain. Can you do this ?

I really can't see where all this is going. Yes numbers and the theory of numbers form an important part of Mathematics. But they are not fundamental concepts, although a numbering system, very different from our own, was probably the earliest maths to 'studied'. Note that the Australian aborigines have only 3 numbers one, two, many. I agree that numbers are very important in applied maths since this often deals with quantities. But what about the rest of Mathematics ? And what about the (physical or engineering) subjects Mathematics cannot tackle ? What about the difference between synthesis and analysis ?

Go on, it's your thread.

What is 'the Language of Mathematics' ? As far as I know it does not extend to miracles.

Torsion is not a form of curvature. Further the direction vectors for both torsion and curvature do not live in the same 'space' as the line itself.