studiot

You are much more likely to be treated seriously if you put some effort into proper English. I am having trouble interpreting your uncompleted sentences. For example what do you mean by the first phrase "be treated seriously" ? or What exactly is your question here? If your question is what I am guessing it is then I recommend you discuss the subject with REg Prescot - he like trying to classify everything as right or wrong - a demonstrably impossible task. As to the process of discovery you seem to be alluding to, this has served Science well many times in the past when anomalies are first noticed , then investigated, then compared and recompared until better and better explanations are developed. For instance the theories of sunken lost continents which were supplanted by continental drift which was supplanted in turn by plate techtonics which will no doubt be at least significantly modified before it is finished or also supplanted in the future.

What do you mean by this? The sphere is a surface, it does not possess one, it is not a solid object. Furthermore it is not conformable with the plane(which is truly two dimensional) What, for instance, are the latitude and longitude of the poles?

Science and Truth, like love, are many splendored things. So you do not do their justice with your analysis, indeed you do their causes a disservice. Knowledge and belief, (why was belief omitted from the original list?) are more focused, but still retain a degree of multifaceted character. I know ( at the level of meaning of know) that there are flying insects called butterflies. This is pure knowledge) I further know there are other similar flying insects called moths. (This is actually the overlap area between belief and knowledge) I do not know the difference but I believe there is one. (This is pure belief) And all of this is part of Science. Your overlimiting proposition thesis is more like the man who sees a tree and says "Forest". Or the man who says "There are three platonic solids" Are these right or are these wrong?

As I thought. What axioms do my great great great grandfather's model "Cutty Sark" satisfy? When they built the real Cutty Sark they worked from a pattern on the lofting floor which controlled the shape so the real ship could be said to be the model and the pattern the axioms by your definition. I can't see where this might be but I will look back at what I did say and if I misspoke then I apologise and thank you for pointing it out.

It is an interesting idea. Have you any information on the economics for isntance how much hdrogen you might obtain for how much limewater injection? How would you trap and store the hydrogen, bearing in mind it is the second smallest and lightest of all gases?

As an aside, heat is not molecular motion, so if this is an accurate paraphrase, Kripke is confused. But then, getting the science wrong is one of the potential problems of philosophers commenting on science. Hi Reg, Just to add to swansont's comment Have you ever heard of heavy water? Some say that Philosophers tend to over complicate but I suggest they often try to over simplify but postulating categories and trying to pigeonhole everything into them regardless of what turns up.

I believe I said there was a difficulty with this one, but forgot to say what thet difficulty was. The double negative means that if there is no solution, something prevents there being one. This means there must be a contradiction with that something. Yes my example was different from yours since there is an integer solution to your example diophantine equation (x=-3, y = -1) as I said. So displaying a solution prevents there not being one. As a matter of interest my view of number systems is that they arise with the expansion of the search for solutions to ever more complicated equations. Usually the old number system is a subset of the new one, and I have only appended one additional number to the rationals to form my new one. "If an assumption would otherwise be inconsistent with the other axioms, making it an axiom leads to an inconsistent system." That is what I believe I said, yes. In what way is my concept of an axiom flawed? A model is a copy of some part of the original. A blueprint to work from (which I think is your meaning) is better replaced with another word - many have been used - pattern - framework - foundation. This is, of course, my opinion and you are entitled to yours, but if this is another example of a professional in one corner taking a word with an already well documented meaning by its originating professionals, and giving it another almost opposite one, then more confusion will surely arise. I am leaving for the coast, early tomorrow morning so I don't think I will have much input for a week or so.

Yes indeed all true. But the next one is a difficulty. Suppose I wanted the solution to the equation [math]\left( {{x^2} - 2 = 0:x \in Q} \right)[/math] This does not exist, because a solution to this equation is not in Q. However I could assert that there exists a solution called a humperdink whcih is in the extended rationals, and so long as I didn't contravene any of its properties I could work with my humperdink. In fact there is a solution in the extended rational field [math]\left( {r:r = a + b\sqrt {2:} a,b \in Q} \right)[/math]

I was really interested in how your diphantine example was a counter example. I am distinguishing between rules and axioms and properties. Don't forget I specified that As you know I can't define a humperdink as a member of the set of Diophantine equations with no solutions.

Inconsistent with. Do all the other axioms you are referring to, taken together, preclude the existance of an infinite set? I disagree with your use of the word model. Models follow they don't precede.

If instead of asserting an axiom, you started Assume a set S exists such that.....etc or Let there be a set S such that..... or What would such a statement be in conflict with? A similar question to taeto

Look again at my explanation of "There exists." It is not necessary to have a rule which states explicitly there exists and humperdink, if the existance of a humperdink does not conflict with any establish rules or axioms. You can define a humperdink, along with its properties, to be wanything you want, so long as there is no conflict. You can then correctly assert "there exists a humperdink" Your problem comes when someone comes along and points out that there is, in fact, a conflict as I did with my simple example of a non negative D. My compliments to Strange for some incisive thinking. +1

Ah, progress at last. "I use calculus" So you are perhaps hoping to 'streamline' the presentation of derivatives. ? Given this I should not have simply substituted dx for h as suggested in your original post. "All that I am trying to do" is pitch my response at the appropriate level. When someone is first starting calculus there is a big difficulty of presentation because they will also be fresh to the other subjects included in analysis, ie coordinate geometry, sequences and series, convergence and so on. So you have to give less than rigourous explanation, which is leaky around the edges yet facilitates progress. This is what teachers do. Then they revist the subject applying more rigour, perhaps several times in all. The other part of the introduction provides for a way to establish some basic derivatives to work with and some basic combing rules for more complicated derivatives. That is the purpose of the definitions you are employing. If you don't have means of calculating some derivatives, there is no point in studying them at all. Most calculus courses then proceed to concentrate on differentiation of more and more complicated (algebraic) expressions, by manipulating the formulae developed form 'first principles' before attacking the more difficult underlying theory. This is why I asked you where you fit into this developing process of understanding calculus. Now one notation for the derivative is dy/dx, but this is not a fraction like 1/5 or 237/390 or whatever, but it is a definite number, if it exists at all. It is not a variable for a given value of x (and y). Following the simple idea of approaching something as closely as we please that is developed in the simple introduction to sequences and series that accompanies the simple introduction to calculus. So we can easily conceive of a real fraction being a ratio of two (small) quantities that are variables. Because they are variables we can allow them to vary in a controlled manner. This controlled manner forces the ratio of these variables to approach, ever more closely, the actual value of the derivative at (x) Which brings us to your observation (again correct) I am not sure what you mean by 'order', but I will give a formal definition in a moment. First I will say (as I did before) there are difficulties lying in wait down the line if you substitute dx for h. These become more apparent when you tackle partial differentiation and what is called the 'total derivative' and also before that what americans call the chain rule and we call function of a function. (I don't know if you know about these because you haven't said) I will also stick with Leibnitz dy/dx notation since it is easier down that line. Finally I call these small quantities the infinitesimals [math]{\delta x}[/math] and [math]{\delta y}[/math] for reasons which will become apparent shortly. The ratio of two small quantities which are both indefinitely small may be one of 1) Finite 2) Indefinitely small 3) Indefinitely large This leads to the notion of 'the order of small quantities'. 1) Two variables, p and q, each of which tends to a limit of zero by themselves are said to be indefinitely small quantities or infinitesimals of the same order if the ratio q/p is finite. 2) If this ratio tends to zero (becomes indefinitely smaller and smaller as p and q become smaller and smaller), q is said to be an infinitesimal of an order higher than p. 3) If this ratio tends to infinity (becomes indefinitely larger and larger as p and q become larger and larger), q is said to be an infinitesimal of lower order than p. 4) If the ratio q/p2 is finite then q is said to be an infinitesimal of the second order, if p is taken as an infinitesimal of the first order. And so on. Exmples of this are:- If r is the radius of a (very small) sphere, so that r may be considered an infinitesimal then The surface area of that sphere will be an infinitesimal of the second order The volume of that sphere will be an infinitesimal of the third order. OK so back to calculus Remembering that dy/dx is a fixed number we wish our formula to approach or get ever closer to Expressing the (small) change in y as a function of a (small) change in x [math]\delta y = \frac{{dy}}{{dx}}\delta x + k\delta x[/math] where k is some quantity that tends to zero as [math]\delta x[/math] tends to zero. So that Now k, [math]\delta x[/math] and [math]\delta y[/math] are all infinitesimals operating under the rules I have just given. If [math]\delta x[/math] and [math]\delta y[/math] are first order infinitesimals, then the product k [math]\delta x[/math] Is an infinitesimal of higher order So [math]\delta y \approx \frac{{dy}}{{dx}}\delta x[/math] or [math]\frac{{\delta y}}{{\delta x}}[/math] approaches [math]\frac{{dy}}{{dx}}[/math] more and more closely the smaller [math]\delta x[/math] becomes. This is the full presentation I did not have time for the other night, so I told you where you could find it. And also shows why I should have followed my own invocation to distinguish between dx and delta_x. and so should not have substituted dx for h, which is another symbol used instead of delta_x.

wtf I know there have been some bad storms in America recently, so I can only assume that you have been affected by the same type of Kansas tornado that affected Dorothy. I can't see that your last tirade of unreasoning outpouring has done this thread any good. So I can only suggest we postpone any further discussion until you return to your normal self.

No wonder I couldn't measure it using my multimeter. I have been trying for the last hour but I had it set to frequency and wavelength. Yes, the point is that there are many connections between different quantities in this or any other universe. So even a small change to one would affect lots of things. If c had a different value in another universe then the wavelength of that light would be different.

Glad you agree on this. This seems somewhat inflexible considering I have already posted an extract from a well recognised textbook using exactly that notation. I Note that (from a straw poll) American practice most often uses Upper case delta, whilst British and European practice uses lower case delta for the same entity. I advocate the reservation of the upper case delta for finite differences since these preceded the calculus of real variables in history and are still in important use today. Hence the use of lower case delta for something entirely different. How would you write the formulae for forward, backward and divided differences, Gregory's formula etc? In my opinion this sort of thing is quite enough use (and very good and compact use it is) for upper case deltas

Yes that would be an excellent clarification, especially in relation to your good point about a never ending trail.

Professor Cox is quite a prolific showman, so how about a more detailed reference to what he was talking about? I am particularly interested in a multimeter than can measure the speed of light.

So what was wrong with my explanation?

I have always understood the definitionof an elementary particle to be Elementary particles have no model or substructure, that is the nature of the beast. Models are made in terms of something more fundamental that that which is being modelled. So a model of an elemnetary particle would of necessity be non particulate. QFT goes a long way towards this goal.

Yes. You are asking a mathematical question of existence, not a physical one. The mathematical statement there exists a set G such that........... means that set G and its conditions do not conflict with each other or the rules employed. So the statement there exists a number C, such that C = 6 + 6 is consistent with both number theory and standard arithmetic rules. It does not mean that I can go down to the supermarket and buy a dozen 'number Cs'. Compare this with the mathematical statement There exists a positive number D such that D = 6 - 12 D does not exist since there is a contradiction of conditions, although there is not contradiction of the rules of arithmetic. In other words I can point to a contradiction (or rule) that forbids the existence of D.

So someone was wrong? Upper case delta, (followed by a referent) is the difference between two fixed values (proabably in a table) of that referent. So [math]{\rm{\Delta x}}\;{\rm{means}}\;\left( {{{\rm{x}}_{\rm{n}}}{\rm{ - }}{{\rm{x}}_{{\rm{n - 1}}}}} \right)[/math] all three are fixed or constant. Lower case delta followed by a referent means an arbitrarily small increment in that referent. To be arbitrarily small it must be a variable. As a variable it is an increment in the referent, which is also a variable. So [math]\delta x[/math] is an arbitrarily small increment in the variable x. To create the limiting process you refer to it is only necessary to append the following improper numbers to define a derivative [math] + \infty [/math] [math] - \infty [/math] Cantor was the first to operate this way (Math Analen 1872, p128) A very long time after Newton.

So your definition means a fixed value then? That is the only material difference from mine. Since it it fixed, how can an infinitesimal tend to zero or just get smaller and smaller? ?

I am waiting for your definition of an infinitesimal, that I asked for a while back. I offered my best one. Going to a grammar school I did Latin. (That was the English definition of a grammar school) evanescent increments was not translated by 'some historian'. It was part of a very famous attack on Newton by the Church of his day. Yes I believe I said something similar in my first post. I don't know if you mean [math]\Delta [/math] ? But this is connected to Newton thinking like a physicist (Which he was in all but name) William Playfair, the accredited inventor of line graphs, pie charts etc was just being born when Newton was in his grave twenty odd years. Newton and his contemporaries worked from tabulations. Newton developed an advanced calculus of finite differences, characterised by the use of upper case delta to denote a finite difference. These were fixed values and most decidedly not infinitesimal; they were (and still are) sometimes quite large in value. Newton used these to fill in or interpolate gaps in his tables, but I thought you knew all this. So it is not a great step from big(ish) differences to small differences characterised by lower case delta, and thence to differences as small as desired. Later mathematicians extended this idea to the 'epsilon - delta' construction you will find in many modern higher level texts on analysis; again I'm sure you already know this.

I have never tried to read Leibnitz. What do you understand by evanescent increments? https://evanescentincrements.wordpress.com/about/ Maybe I'm wrong about some detail but I don't know where the attribution of the ratio being the fluxion has come from, my sources seem to clearly indicate that Newton considered [math]{\delta x}[/math] and [math]{\delta y}[/math] as fluxions.