Wednesday, November 26, 2008

...on classical Greek mathematics, or logarithms for use on yardsticks

I wrote:

On a thread on geocentrism vs. heliocentrism, I and VoP differed on whether there are any NUMBERS other than NATURAL NUMBERS: whole numbers from one and potentially ad infinitum, not as if there were any infinite number, nor as if there were not any number that is really the greatest number in the universe, but because there is no particular arithmetic reason known to us why it should be the greatest, nor do we know how great it is: it is only actually greatest, while numerically there remains a potency to greater numbers, without limit.

I said nay, nothing else is numbers, fractions are ratios rather than numbers, pi and sqrt(2) are geometric incommensurable proportions rather than numbers or ratios, negative numbers are misnomers for natural numbers of "negative" or negated things, and I stick to it. However Voice of Principle challenged me on logarithms. According to my definition of numbers, he said logarithms would not exist.

In a sense, I do not think they do, but only in a sense. There is no arithmetical 10log of 2, because no potency of two exactly equals a potency of 10, just as there is no arithmetical sqrt(2), because 2 is not a sq #.

This said, I must go on to say that my classical Greek definition of numbers does not stop me from seeing what the 10log of 2 is, in so far as in any sense it is. Rather, I see it more clearly.

Even earlier, I had a hunch to figure out the 10logs anew, but in fractions expressed in duodecimal fractions rather than in decimals. But it is a pretty hard work to figure out the ninth root of ten (cubic root of cubic root) and put it in the fourth potency (square of square) just to get at the value for 10log : 4/9.

A few days ago, I started anew.

10 to 1/3 must be more than two, because 8 to 1/3 is two. 10 to 1/2 must be more than three, because 9 to 1/2 is three...

Before going into my preliminary results, a little terminology:
  • 1' (foot)=12" (inches)
  • 1"=12'" (lines)
  • 1'"=12 "" (points)
  • 1'=12"=144'"=1728""
The smallest unit points is not a current English unit, but the French pié du roi (1/6 of Charlemagne's body length) is subdivided ultimately into 1728 points du roi - analogically I speak of points as the 1728th part of any foot-measure.

The cube (3d potency) of ten (=1000) is less than the tenth potency of 2 (=1024). So 3:10 is less than the 10log of 2.
  • 144:10=14 2/5
  • 14 2/5*3=42 6/5=43 1/5
If 1=1', 43'"1/5 is the lower limit of 10log of 2.

Next line is of course 44'"=44:144=11:36. And the 11th potency of ten - 100,000,000,000 - is greater than the 36th potency of 2 - 68,719,426,736 - so 44'" is upper limit for 10log of 2 in the first approximation.

4 is the square of 2, and to find the 2d potency of a number, you multiply the logarithm with two, according to general rule that adding logarithms means multiplying corresponding numbers.

The 10log of four (=10log of 2*2) would thus be between 86'" and 88'". 87'" come in between. 87:144=29:48. Is that more or less than the 10log of 4?

More. Which means that as 88'" are reduced to 87'" as upper limit for 10log of 4, so the half must be reduced from 44'" to 43'" 6"" as upper limit for 10log of 2.

To get the 10log of 5, subtract 10log of 2 from 1 (=10log of 10). To get that of 25, add it to itself. To get that of 2.5 subtract 1' from 10log of 25. Add 10log of 5 and subtract another 1', and you have the 10log of 1.25 - and when it comes to the 10log of 3 I jumped straight onto the 10log of 9: 1020 is less than 921, but 1021 is more than 922. So the 10log of 9 must be between 20:21 and 21:22. If it is expressed in duodecimals it is easier to halve to get the 10log of 3.

My results so far:

10log of 2
> 0' 3" 7'" 2""
< 0' 3" 7'" 6""

10log of 4
> 0' 7" 2'"
< 0' 7" 3'"

10log of 8
129 - 131'"

10log of 5
> 0' 8" 4'" 6""
< 0' 8" 5'"

of 25
> 1' 4" 9'"
< 1' 4" 10'"

of 1.25
> 0' 1" 1'"
< 0' 1" 3'"

10log of 9
> 0' 11" 5'" 1""5/7
< 0' 11" 5'" 5""5/11

and of 3
> 0' 5" 8'" 6""6/7
< 0' 5" 8'" 8""8/11

Always presuming that 1=1'.

Now, if VoP would pls check the accuracy of my most accurate logarithms (2, 3, 4, 5, 9), he may see for himself whether my clinging to Classic maths has stopped me from understanding logarithms!

Now for a theoretical definition:

  • a logarithm is not a number, but EITHER a ratio between the (whole number!) potencies of two numbers, the base and the number whose logarithm it is, so that
  • if basea=numberb
  • then the logarithm is a:b
  • OR a geometric irrational proportion that can only be approximated to above
  • and furthermore expressed in either case as fractions of an arbitrary length unit, so as to compare with real or virtual counting slides (is that what you call them?) the potencies of numbers, so that multiplication of numbers can by succesful fiction be expressed as addition of potencies and divison by subtraction, potencies by multiplication, roots by division.

No need to dub logarithms numbers in order to understand them, then!

Hans Georg Lundahl

Continued:The fact that there is no such a thing as a 10log of two is also proven by the fact, that the closer approximations to its value - the LESS they have of the definition of logarithm, i e ratio between exponents of EQUAL potencies.

In order to prove this, consider that 87:288=29:96 is a closer upper limit than 44:144=88:288=11:36.

Now, will the potencies 1029 and 296 be more or less equal to each other, than 1011 and 236?

Two96 is
79,228 quadrillions
162,514 trillions
55,647 billions
658,951 millions

which is more than 20 quadrillions off the 100,000 quadrllions that form the potency 1029.

Clearly this difference is greater than not just the difference between 1011 and 236, but even greater than any of the number involved in that real inequality and nearest possible equality.

Oh, yes - when I had taken the sweet trouble (like a crosswordpuzzle) of calculating 296 in a few grid systems, I found it quite as worthwile to go up a few potencies of two by doubling.

31:103 is a closer lower limit (closer than 3:10), because
10 quintillions
141,204 quadrillions
801,799 trillions
122,900 billions
345,849 millions

is greater than 10 quintillions.And the upper limit can be drawn down to 32:106.

The new approximations are:
0' 3" 7'" 4""8/103
0' 3" 7'" 5""35/53

Which I found out in proving that what is thus approximated can be infinitely approximated and never reached because it doesn't exist.

Résumé of mathematic debates with Voice Of Principle:

  • a) He attacks my argument against the regress to infinity "infinity cannot be passed through" as having been refuted by modern mathematical understanding of infinity, also he attacks my logic on logic thread by claiming there is reason that goes counter to logic and is still true
  • b) I answer that every number is finite, a multiple of one, and that every number is rational
  • c) He counters with saying that Greek math's thought so, but PI and sqrt/2, being irrational, disprove this
  • d) I answer that I know very well that PI and "sqrt/2" are irrational, it is the number part of their categorisation I disagree with, since they aren't numbers but size relations
  • e) message disappears
  • f) when I repeat the point, VoP claims my limited understanding of number cuts me off from understanding the great new "discoveries" of math's since Newton
  • g) On a thread on geocentrism/heliocentrism, Rita claims the main argument for heliocentrism is that Copernican hypothesis of Universe makes accurate calculations of planetary movements possible
  • h) I counter saying that mathematic fictions can make calculations easier without being true to mathematic realities and give as example the fictitious negative rule of squares (a - b)sq = asq - 2ab + bsq, proving this is geometrical nonsense if taken to the letter, step by step, as contrasted with real rule (a - b)sq = asq - bsq - 2b(a - b), which is true to geometry, involves no supposition of negative numbers existing, but is less handy
  • i) VoP claims I misrepresent algebra and claims it doesn't involve any fiction, repeating that my limited understanding of mathematics cuts me off from many great discoveries
  • j) I disprove both his points by this thread, calculating the 10logarithms of 2, 3, 4, 5, 8, 9, 25 and 1.25, especially refining the logarithm of 2, while saying that it is not a number and what it really is: a relation, and, since exponents must be whole numbers, a fictitious relation between exponents of 10 and 2 when their powers equal - which Eratesthenes has proved they never do. To substantiate my claim of calculating the log of 2, I show my calculations in part and give the values in duodecimal fractions, corresponding best both to my old dream of making counting slides on a yardstick and to my calculations - and leave it to VoP to convert the duodecimals into decimals to check my accuracy
  • k) VoP does not answer, but AbbyLeever, who has not followed my debates does, repeating VoP's misundestanding of my arguments.If he had been a zen buddhist, I think he might have understood my mathematics better - not that that would have saved his soul of course, but it would have been more stimulating on this board.

Hans Georg Lundahl