studiot

Why not ? They have a relative velocity, isn't that a physical interaction between two distinct entities ?

How about the Lorenz transformations ? They are equivalence relations. https://deepblue.lib.umich.edu/bitstream/handle/2027.42/152115/A_novel_equivalence_relation_in_relativity.pdf?sequence=1&isAllowed=y

Can it ? Let us try some examples. Compare the distance between 1 and 2 and between 4 and 5 by this definition. Distance 1,2 = √(12 + 22) = (1 +4) √5 Distance 4,5 = √(42 + 52) = (16 + 25) = √41 Is this difference acceptable ? If you read my Wiki reference on Metric spaces you would see that there are three different common definitions of distance and the euclidian one is not appropriate here. I have already mentioned the method of Dedikind. This used to be the preferred route to understanding why we can keep chopping up the distance between any two real numbers. The old way was to start with counting numbers, (whose distance betweeen is not infinitely divisible) show why these are not enough ie there are more whole number than counting numbers to get the integers Then progress to show that there rational numbers give us still more numbers the distance between becoming infinitely divisible. But also that there are gaps in between which means that the rational numbers are not continuous. Finally we come to the real numbers which are continuous in that there are no gaps at all such that the real numbers possess the property we call completeness as well as being infinitely divisible like the rationals which are not complete. Being complete and without gaps means that that there are no further numbers to acount for.

Further to my last post, this topic makes me think of the mathematical idea of equivalence relations. The idea has wider application than just maths; many physical interactions are relations, but not equivalence relations, in that one or more of reflexivity, symmetry or transitivity ar not satisfied. https://en.wikipedia.org/wiki/Equivalence_relation

Do we not also recognise the notion of self-interaction or self-activation for some phenomena in Physics ?

Your answer is correct so it is a pity your reasoning is incorrect. furthermore numbers don't 'know' anything. Also I don't agree that you claim about extra drops of water follow from your premise. Surely the answer will be yes.

This statement I can agree with. So here is a simple observation. The ancient Greeks realised that If r is a positive number, real or otherwise, then r+1 is a larger number, and (r+1)+1 is larger still and so on. So there is no largest number. Now we also know that if we take reciprocals 1/r is greater than 1/(r+1) which is greater than 1/((r+1)+1) and so on. Since there is no largest number there can be no smallest number either. Sweet dreams everybody.

A very reasonable question. Have you thought about possible tests ? Interestingly this is made possible because of the finite speed of light. We can look back over much longer timescales than our own lifetimes or even our civilisations'. So we can compare the configurations of matter over very long timescales and we find that the further back in time we look the smaller the gaps are and the average density is diminishing, when directly comparing one with the other so we can say with confidence that the correct interpretation is expansion not contraction.

Subject to the comment Genady has already made about Reiman integrals, yes you can and The method is nothing like you seem to envisage and nothing to do with Reiman integrals. First you you should study this article carefully. https://en.wikipedia.org/wiki/Metric_space What you are seeking is the method of partitioning a set so that every partition contains exactly one member of the set and there are no empty partitions. This is called the method of Dedekind cuts. But it does not lead to a smallest real number,

One or two debatable points there but your heart is in the right place. +1 for encouragement @Boltzmannbrain Whilst @Genady is doing such a good job offering straightforward textbook details I think it is worth looking deeper into the number line concept. I will readily grant you that the number line is a very attractive way of presenting aspects of the properties of numbers. But what we are really doing is as in fig1 putting numbers in one-to-one correspondence with a straight line and hoping that other properties are then transferable. But also as in fig 1 it is easy to see that the actual length of the line is irrelevant - it works for all straight lines, regardless of theier length. Radical conclusion : You can establish one-to-one correspondence for the whole set of numbers with the shortest possible straight line, as well as the longest possible. Question : How does this affect you proposed distance function ? Surely the distances apart of any two numbers will depend upon the length of you line. But who says the line must be straight ? Fig 2 shows how to use a spiral to place the numbers. This has the interesting property of being able to cover the entire plane. Question : How does this affect you proposed distance function ? As can be seen in fig 2 - the numbers 2,4 and 6 are all equally close to the number 1, whereas the numbers 3 and 5 are further away. But does the line have to have a definite geoemtric shape ? So far I have discussed straight and spiral. Surely all that is required is that it must not cross itself. Question : How does this affect you proposed distance function ? Can you see why it must not cross ? Fig3 shows one such semi random arrangement. There are an infinite count of such non crossing arangements. What about the numbers themselves. There is nothing in basic set theory to say that they have to appear in the set in any particular configuration. Fig 4 shows a random positioning of the numbers which is perfectly permissible. Question : How does this affect you proposed distance function ? Clearly 4 is the closest number to 1. So that you can have them is value order you have to add further structure to your set of numbers so that you can introduce a 'well ordering' of the set. Your thoughts on this ?

Great science. +1

Thanks for the extra info +1

Sounds like someone is trying to pull the metaphorical atoms of my metaphorical leg to me. Both your statements 1 and 2 are false so I suggest you start again with a proper explanation of what you are trying to say.

If you want to discuss numbers (or anything else) mathematically you need to only use properties that numbers have, not introduce new properties they do not have more especially not when you name those properties something whuch already has a specific mathematical definition as something other than your new properties. I won't both again to say that numbers are not lines and lines are not numbers. You will not find a single number as an element of a set of lines or even the set of all lines. Similarly you will not find a single line as an element of a set of numbers or even the set of all numbers. You keep mentioning a well define property of lines notably distance. How do you define distance ? Numbers do not have this property , they have the property of > < or =, and you must use these to couch you arguments in. Equally, lines do not have the property of + - x or / as well defined operations nor do they follow the normal rules of arithmetic. The conventional way of presenting the notion of a limit as getting closer and closer to something is called the epsilon-delta argument, which involves the normal arithmetic operations along with the less than, greater than or equals notation. Have you heard of this ? I can show you this if you are prepared to listen Are you prepared to listen becasue at the moment you do not seem to be. @Genady has offered you an alternative method via sequences, you don't seem interested in these either. I should warn you that this only works with certain types of sequence called null or Cauchy sequences. They don't. But there is nothing wrong with your comments. A finite range is just the difference between two selected numbers.

The difficulty with answering your queries is how much you do or do not remember from the past. What you need to know here is this minor theorem. Every real number has a decimal representation ( note it is not necessarily unique) So any decimal you can write will be some real number. So is 0.0 repeating a real number. Yes But is 0.0 repeating the smallest real number ? No -1 < 0 So to your worry over an infinite count of zeros after the decimal point, so what ? Pi is well known as and infinite string of digits, as is 0.3 recurring. The process never ends that is what infinity means! Since it never ends, there is no last digit in an infinite string. So the situation you envisage cannot arise. No matter how many noughts you have after the decimal point you can always add another one. This is a fine example of using the wrong words, resulting in nonsense. As is this leading to a statement that is just plain wrong Only some aspects can be thought of geometrically, and only in some circumstances. And distance is not really one of them. Consider this Every number has a square. (theorem) So let us lay out a line of numbers against a line of squares 1,2,3,4,5,6,7,8,9 1,4,,9,16,25,36,49,64,81 Every number on the first line is in one-to-one correspondence with a number on the second line. but the 'lines' have quite a different character.

Well I have yet to see anyone in England that can make a half ways decent paprikash or goulash - or even know the difference - except my late mother. I only wish I could do better than come near her standard. And then her bannocks. I have never been able to replicate those.

Sometimes it is just a question of getting the right words as using popular general ones in their non scientific sense can lead to misunderstandings. 🙂

I only wish that were so. Because I find it really galling to be served something at 5 to 10 times the price of my home cooked and think "I could do much better".. Most folks who sit at my table agree. And I have never poisoned anyone, unlike some self styled high class establishments. When I was first at uni, one of my flatmates came from South Shields. He had a copy of the local rag sporting the headline "Chinese Restaurant Steeps Peas in Toilet." Go figure

Exactly. +1

This simple scheme might help. Which way do you want the maths to go more or less complicated? Which of these real numbers is the next number after 1 ? 1.1 1.01 1.001 1.0001 1.0001 1.00001 1.000001 etc

The word point is much overused. I debated with myself how to avoid it as far a possible to avoid confusion. Totally agreed. But I do not like the use of the word end in relation to lines as it can be imprecise. An open interval has no end. I am sorry you missed my main point (there is that word again, but with a different meaning this time) So it means that my explanation was not good enough so I will try to do better. But it is difficult to know what you know as you obviously have met the idea of open and closed intervals before since you used an alternative notation in your responses. I carefully avoided [0,2) etc because it is easy to overlook which bracket is which and you can never be sure whether the writer meant it or not. The reversed square bracket stand out, don't you think ? Also curved brackets are used to denote sets. Anyway you clearly understand that part. My main point was that lines and numbers are not the same, although they have some properties in common, which allows one to exemplify the other when only the common properties are of concern. But sets of numbers have lots of other properties where they cannot be represented as lines. So mathematicians seek more general approaches. If there is a number that is greater than any other number is the set then the set is bounded. In fact we say it is bounded above and can say (similarly it is bounded below if there is a number less than any other number in the set.) So the set (1, 2, 3, 4, 5, 6) is bounded above by the numbers 6, 7, 8, 9, 10... We call any of these an upper bound. Also the upper bound may be an element of the set or it may not. When the upper bound is an element we call it the maximum of the set. Now there is a theorem, which I will not prove, called the least upper bound theorem. "If any set of numbers is bounded above it has a least upper bound" In our example 6 is the least upper bound of our set and is also the maximum. But our set is also finite so it is easy to see this. Finite sets means that the count of elements is finite: Infinite sets have an infinite count of elements. The boundedness theorems apply to finite and to infinite sets, (But not the max and min) Infinite sets can also be bounded. The set of all the elements of the negative exponential e-x, from x=0 to x = ∞ is bounded above by 1 and bounded below by 0, although the set is infinite becasue the count of x values is infinite. Note the x = ∞ 'end' is never reached or as I prefer there is no right hand end to this line. So this line has no minimum. The left hand end depends whether we include or exclude x = 0 in the set (closed or open ) If we include x = 0 then the upper bound of 1 is also the maximum, But if we exclude it then again the upper bound is never reached and the set has no maximum either.

An abbreviation of british slang expression, common from about 1920s to 1970s. Right ho. Meaning I acknowledge what you have said and will carry on on that basis.

Very quickly before I head for the bank Part 3 of Callen Thermodynamics and Thermostatistics pages 455 - 471 Callen goes through modern interpretation of the quantum angular momentum rules via Noethers theorem (symmetries), Goldstone's theorem (broken symmetries) and gauge symmetries in relation to accessible and inaccessible microstates of a system and the state space of a system. The quantum rules forbid certain transfers of angular momentum which leaves ortho and para hydrogen as essentially different gases in a mixture.

Glad it worked out for you! It was late and I was rushed last night. One of the very first books I bought at university back in 1968 was Denbigh's Chemical Equilibrium. He gives a much wider treatment of these effects both from a fugacity/Chemical Potential point of view (P1`22 - 132) and also form a statistical/calorimetric entropic point of view. Here are some pages of the latter which may be of interest since they cover a much wider range of gases that just hydrogen. The note at the bottom of p420 refers to Glasstone - an earlier work from 1940 - on page 588 My 1951 second ed has that material at the beginning pages 94 - 97 entitled Ortho and Para States. This may be more accessible for you as it is american. In deference to your opthalmology I have increased the scan deoth, although it means a larger file size. Yes I agree, the standard chemist's method of avoiding mechanical mechanisms is to go the least energy/ least action route. Another good reference is Wilson Thermodynamics and Statistical Mechanics 1966 Chapter 6 Specific Heats p 139ff 6.12 The Rotational Specific Heat of Hydrogen