You came along just as I was being exasperated. And yes, you are right, that as far as the UNDERSTANDING of modern things is concerned, I am a pre-Enlightenment scholastic, a thomist who would out-thomist Maritain, as it were.
You are right that "the devil" if that is what you like to call it, is in the definition.
The thing is, a few months ago, you saw that I stuck to another definition of number, the old one, and challenged me that it hampered me in understanding inter alia logarithms (not to mention "integral calculus" which has hardly anything beyond the name in common with calculi and the numeri they count). My first message on this thread was my answer to that precise challenge. And I expressed the values in inches lines points
- because that way I was neither helped nor hampered by knowing that 10log of 2 is approx 0.301
- because it stressed that fractions to me are not numbers, that are expressed in exponents of 10, but values of some continous greatness, like length, which is measured in duodecimal fractions on the scale relevant for a counting stick, in the old system
- because I like the old system more than the sham scientific, pseudoexact metre system: in continuous quantity the unit is arbitrary and the measure of all things is (6') a man
- because the duodecimal fractions are great for expressing the Fibonacci numbers 89, 144, 233, 377: count as many lines('") and you have good approximations of golden ratio on a handy scale.
Another clarification: when it comes to the RIGHTFUL USE of modern technology, I am ethically a Luddite. If you use the flying shuttle or Spinning Jenny for your amusement or strictly domestic needs of textile - go ahead. If you use them in Wars, when weavers take up arms, or catastrophes, when blankets and clothes and bandage linen must be quickly produced without economic profit - better still. But if you employ them to replace workers in ordinary, peacetime, commercial production, and to compete others who refuse to do so out of business - smashing is what they deserve. That is why I claim to be proud to be a Luddite.
Hans Georg Lundahl
VoiceOfPrinciple wrote:
<1> Hans, I have done little reading and even less posting on this board for the least several weeks due to an acute lack of time. By chance I noticed this particular thread last night and read all of its posts. AbbyLeever has done a fine job replying to various points you have raised, so I am going to content myself with a few general observations, rather than repeat her effort to no practical effect.
The devil, as it were, is in the definition. You have defined a number to be a positive mathematical value with a zero fractional part (that is, an integer). Both I and AL would define a number as a mathematical value. Period. A mathematical value is something which can be transformed by a mathematical process such as addition, subtraction, multiplication, division, exponentiation, etc. This is the heart of the difference between your conception of a number and those of modern mathematicians.
Having said that, it is far more interesting to ask why you define a number as you do. Some others on this board refer to you as a Luddite. This is both imprecise and misleading. You are in fact a pre-Enlightenment Scholastic. I suspect you regard the Enlightenment as a great misfortune for Western Civilization, since it called into doubt fundamental religious doctrines which you hold near and dear. Anything associated with, or derived from, the Enlightenment (such as modern mathematics) is to you suspect. Hence your insistence on using an historically interesting, but limited, view of the mathematical realm.
I cannot help but close out this post by pointing out a certain irony that may have eluded you. Modern mathematics, at least through the middle years of the 20th century, is almost exclusively a product of Western Civilization. Many of its greatest contributors were practicing Christians. Classical mathematics, to which you are committed, on the other hand, was developed by a pre-christian culture composed of Pagans. It's not just politics that makes strange bedfellows. <1>
AbbyLeever wrote:
Are you done? I believe I am vindicated by VOR in spite of your bombastic retoric to the contrary.
It has been a good laugh.
Do you know how a ship leaves harbor? It raises it's anchor.
--
Vindicated from misunderstanding? NO way. VOP knew from previous encounters how I define numbers, but you should have grasped it instead of accusing me the third time of doing what I didn't. All the while ignoring what I did, although I had said so on the very first post: vindicated the old definitions against VOP's claim they exclude me from knowledge of logarithms. And though he hasn't said so, his silence eloquently tells me he cannot keep that charge up. Holding logarithms to be not at all numbers, but ideally ratios of exponents, when powers of base equal powers of the antilogarithm, most often approximations to that, there being no equal powers, I have given an accurate value for 10log of 2, in duodecimals, my final approximation being 0' 3" 7'" 4"", giving account for my way to it and thereby showing I have not just plagiarised the known value 0.301 - to which it "incidentally" approximates on the lower edge. How did I do it, unless I understand logarithms? How do I understand logarithms, if holding to the old definitions of number, size, relations rational or otherwise as different things rather than lumping all together as "numbers" stop a man from understanding logarithms? Actually, before I thoroughly got the ancient idea into my head, that number means integer, and therefore that everything counted rather than measured, including exponentiation, must be whole numbered, I could not make proper calcualtions of logarithms. Now I know they cannot be in the full sense calculated, I can make them with sufficient accuracy. For the value given for 10log of two, the errors, where powers of two would coincide with powers of ten (which of course they cannot, which is the reason why 2 has no real logarithm) will occur when the counting slides extend to about ten yards: a tolerable error for a one yard pair of counting slides, I'd say.
Hans Georg Lundahl
AbbyLeever wrote:
I am thinking of a number that can be approximated by the number 5
It can also be approximated by taking any number in existence and finding the number that is 1/2 way between it and the number 5. The more you do this the closer the approximation comes.
What number am I thinking of?
A - if it can only be approximated, it is not a number, but a size relation, rational or not.
B - Do you actually mean that the greater numbers you take and get the half distance between it and five, the better the approximation? Or that you call that half distance a number and halve the distance again and again and again? Or that you average the half distances without repeating the halving?
But most important of all: how do you know that you have really taken EVERY number in existance into account? Obviously you cannot.
C - If you are thinking of 5, which you cannot unless the halving is continually repeated, why approximate what you can have exactly?
D - if the halving of the distance is to be done on a surface with some kind of graphs, it would be a mathematical size or size relation - and hence not necessarily a number nor an arithmetic thing (most assuredly not if it can be only approximated), but rather a geometric thing.
E - would that be "e" that you are thinking of? I have run into a description of it beginning with two, continuing with a + followed by a fraction of which the denominator (I think it is) also contains a + followed by a fraction, of which... why take the trouble to find that approximation, when one can just as well find approximates of 10logs for prime numbers by comparing exponents of base and antilogarithm? Admittedly the 10log of 7 is right now wavering somewhere between 121 and 122'", but I am hoping to get to powers where stricter narrowings may be made by ratio of exponents.
AbbyLeever wrote:
(a) I will let pass for now
(b) in an approximation iteration you take the result of the previous step and feed it inot the algorithm to produce the next result (or don't you do that either) so it is your second statement (and yes you can start with any number)
(c) no guesses, I want your mathmatical answer
(d) the halving is mathmatical, just like every one of your precious fractions
(e) no, keep trying...
H G Lundahl wrote:
[[Starting with ten I get: 8 1/2, 6 3/4, 5 7/8 and it would seem that the lower limit for any number higher than five, higher limit for any number lower than five but a limit never exactly reached would be: five.
This brings me back to my objection:
C - If you are thinking of 5, which you cannot unless the halving is continually repeated, why approximate what you can have exactly?
Hans Georg Lundahl
AbbyLeever wrote:
AbbyLeever wrote:
H G Lundahl wrote:
I am back NOW. Yesterday and the day before, I've had other things to do, and it would seem you have had a really fun time these two days.
None of you noticed a mistake in mathematics I did make, though.
No, it isn't obvious. Unless you were thinking of calling these fractions numbers, it is not to the point of our debate. And though I did make the mistake of ending up in 5 & 7/8 rather than 5 & 5/8 in these steps, I certainly did NOT make the mistake of calling 5 & 5/8 a NUMBER in the full and proper sense of the word.
Five is a number, so is six, and as it is the next number, there are no numbers between them. As 5 & 5/8 is between them, it cannot be a number, nor are 5 and 10 IN THIS CONTEXT, rather it is a question of ratios, though it is till only a question of NUMERIC ratios, common to number and size. 5 & 5/8 is, properly speaking, the ratio 45:8 - above 40:8=5:1, but below 48:8=6:1. A ratio is not a number, unless it has 1 for denominator.
Hans Georg Lundahl
AbbyLeever wrote:
5 is a number
the number 5 is approximated by the series I gave you, but is never reached
the fact that the number is never reached by the approximation is not proof that the number 5 exists, as because of the first statement we agree that it exists
therefor any approximation of a number that does not give a final answer for a number is NOT a proof that the number does not exist
your proof of the existence or non-existence of pi is invalid (note - not proven false, just proven invalid)
and as for the non-existence of numbers between 5 and 6, I just have another set of numbers that are worth twice the original numbers and in it's series then number 11 falls half way between your 5 and 6, exists and is real. It corresponds exactly to 5 and a half, so therefore 5 and a half also exists and is a real number. glad to have you back, and hope you had a good couple of days.
(I just assumed a typo, and it isn't really critical to the argument, like criticising
spelling and grammer is irrelevant to the discussions)
H G Lundahl wrote:
5 is a number in its primary sense - and in that sense the next number is 6, no number coming between.
5 1/2 is certainly a reality, but not a real (meaning true, no issue about whether it belongs to what modern mathematics mistakenly calls "real numbers" being intended) NUMBER in the primary sense of the word.
As you said yourself: its simplest value is 11. Now 11 is not 5 1/2, but rather 5 1/2 * 2. and there is 5 1/2 * 4 = 22. and 5 1/2 * 6 = 33, and so on.
Where 5 1/2 is simply 5 1/2 it is not a number, but a measure, each measure being not a unity in the full sense of the word, but a unity that changes name but not nature by being divided: pizzas, pints of ale, feet and miles, days and years all spring to mind. But there you are not dealing with a number of totally separate items (which is the prerequisite for speaking of number in the full arithmetical sense of the word) but rather of things that in themselves are undivided or accidentally divided: food, drink, size in space, duration in time et c.
When we are speaking of anything that can in the proper sense be called number, namely a number of fully countable things - stones that retain the name or animals or men that change the nature if cut in two - there is no such thing properly speaking as 5 1/2, only 5 1/2 * something else, that else being invariably a multiple of 2 or simply 2.
"5 1/2 *" is not a number: it is a numeric ratio, aka 11:2, which remains identical if the numbers involved are changed in equal proportions: 22:4, 33:6, 44:8 et c. As you will see: every number has a corresponding numeric ratio, but not the other way around. And every numeric ratio can be a size proportion, but some size proportions have no corresponding numeric ratio, because the sizes are incommensurable: whatever arbitrary unit be chosen for measuring exactly one size will be too big or small in any multiple or rational fraction to fit the other: among which you will admit is "sqrt/2" aka diagonal:side of a square, PI, which is short for perimetre:diametre, et c.
Are you finally getting my meaning? Or do you still consider the ancients wrong in saying that every number is a ratio? Or will you after my computation (not calculation in the full sense, since that is only of numbers) of 10log of 2 insist that, although the Greeks were right about it in their definition, they would not get the impressive arts - such as logarithms - of modern maths except by changing the definitions? It was this third point which the thread was all about, after VOP had previously conceded that, according to the ancient definition, PI and sqrt/2 are no numbers.
Hans Georg Lundahl
AbbyLeever wrote:
What I see is that you make a distinction that for me is totally unnecessary and serves no purpose. It is like arguing that there is only one blue color, when the color can be anywhere on an infinitely variable spectrum.
Consider the dividing of a pizza into pieces as separating the molecules into different groups with each molecule whole and accounted for. And then when I have divided it up into individual molecules, I divide it further into groups of atoms each atom whole and accounted for. And then when I have divided it up into individual atoms, I divide it further into groups of particles each particle whole and accounted for. And then when I have divided it up into individual particles, I divide it further into groups of sub-atomic particles each sub-atomic particle whole and accounted for. And now I am in a fine little problem because the sub-atomic particles are constantly changing, becoming different particles or becoming multiple particles or becoming no particles and so on - there is no "1" there is a vibration around the number "1" that is a probability of being "1" but not a necessity.
Ultimately there is no "1" there is a cloud of possible numbers, that could even be 1.5 if caught in the act of changing from "1" to "2".
Except of course that you will not agree with the modern physics portrayal of matter composition, either.
What I see is that "we are limited in our understanding of the universe by our understanding of the universe" - and your understanding is different from mine. For me your mathematics is a subset of mine, while for you my mathematics includes a fantasy outside yours.
H G Lundahl wrote:
My distinction between number and size - namely number being many as opposed to one, something to which one at a time can always be added from the outside, as far as the arithmetic nature of the number is concerned, discrete quantity; and sizes, weights, other measures being rather continuous quantity, infinitely divisible because any division is already potentially inherent in the thing itself, and furthermore my distinction between them as such - the category of quantity - and their relations, the category of relation, to which belong ratios, pi, sqrt/2, logarithms, but also the most straightforward arithmetic relations, like add or subtract five (same relation seen from the two numeric termini) or neither add nor subtract, is part and parcel of scholastic, Aristotelic, common sense understabding of the Universe.
So far you're right. But what subsets of your mathematics does mine lack? Not logarithms on my showing!
HGL
AbbyLeever wrote:
pi, e, the square roots of all numbers, decimals, fractions, the square roots of negative numbers ... just to name a few are included in my mathematics as numbers.
one could argue that pi is what it is because God wanted us to think and not make it easy. at one level all things in the universe are illusion and what we perceive as reality is based on ideas we have not on hard evidence - is the reality of a cup that you just put down still a cup or the image of a cup?
peace.
My answer, so far:
Abby Leever:
Why do you keep repeating the charge that I were regarding pi, "sqrt/2" or of any other number that isn't a sq#, logarithms as illusions? I do not: if they are false numbers they are real size relations of the proportionally constant sort. Just as a false Hector is a real actor (excuse the pun), or more properly as what is for Baudelaire a false musical tone - green - is a real optic colour!
You say I call only a small section of the spectrum of numbers numbers. Not so: I am not limiting colour to only blue, I am excluding C major from colour!
There is for every number except the one that is the greatset, a corresponding arithmetic relation of so many units greater or smaller like for 5 the relations 5 more than or less than, and for every number including the greatest a corresponding relation of the sort called geometric, though it exists already in arithmetic multiplication and division, namely a ratio: like for 5 the ratio 5:1, five times as many as, which has an obverse, 1:5, a fifth as many as.
Note that 4 is 1 more than 3 says exactly the same as 3 is 1 less than 4. Plus and minus are not indeed the same side of the same relation, but the two opposite sides of it, aka equal and opposite relations.
Equally, 10 is 5 times 2 says the same thing as 2 is a 5th of 10. 5* and /5 are equal and opposite proportional relations, aka two sides of the same relation.
Each numeric ratio has ipso facto a corresponding musical ratio, an interval, but there are intervals that are not in the relation of number to unit, like 1:2 or 2:1, a pure octave, or 1:3 or 3:1, a twelfth, but also number to number ratios, like 2:3, a great (pure) fifth, 3:4 a small (pure)fourth, that do not correspond to any one number or any one arithmetic operation, but rather to two at a time, and confusing numbers with ratios is as bad as confusing pitch with interval.
Furthermore each musical ratio has a corresponding size ratio: just as there is 2:3 the pitch or string length (on a monochord), there is 2:3 the size, like length or surface or volume. But there are also proportions that fall between any two ratios and, a fortiori, between any two numbers. Like pi, "sqrts" of nonsq#, golden ratio, possibly e, certainly logarithms, et c. I am not at all denying that what you call real numbers - NB above zero - are real: I am, repeatedly, saying they do not belong to the category of number but to another category which only in part corresponds to it and also has a part not so corresponding: which is why geometry is a greater science than arithmetic.
As for zero, it is not a number: plus minus zero is a name for an arithmetic relation of numeric identity, as far as addition and subtraction are concerned, just as 1* and /1 is the name for that relation of numeric or size identity, as far as proportion is concerned: in usual terms it is called as many as or as great as. In music it is called the pure first, the same pitch. And as for negatives, the negativity resides not in the number, but in the numeric relation, in what direction it is seen from: which disposes of the pretention of there being "numbers smaller than zero" or of them having any roots, whether sq or cb or bisq or other.
There is thus no number line stretching from negative infinity to positive infinity, which disposes of the objection against any proof of God's existence, that is based on the impossibility of the regress into infinity. And no number surface, except in graphs and in Gauss' imagination. And no numeric infinity, which makes it impossible to identify the Infinity of God with any attribute of the manyfold. And defending these common sense proofs of God's existence and transcendence is the whole point of my issue against modern math's - as well as that the confused terminology and the unceasing appeals to broaden ones imagination employed by math's teachers to defend it, make gifted mathematicians fail by failing to understand the explanations as stated. Not that I was a victim - or I hadn't been able to sort this out, perhaps.
Hans Georg Lundahl
