Arithmetic is not Geometry. And Real Arithmetic makes "Real Numbers" unreal. However much you like apple pie, you can never have π apples. However much a tree is rooted, between 1 tree and 2 trees, there is no such thing as sqrt(2) trees.
This is a good refresher of remembering what we really know even in Number Theory. Someone brought up logarithms, and I finally, years later, came up with a model for logarithms, which was obviously not meant to replace the logarithms we have, but I used a different than usual way of expressing logarithms to prove I hadn't cheated by simply looking at a logarithm table, that my understanding of what logarithms actually are in number theory actually had allowed me to find some logarithms.
This time, I'll go with the known value 0.301 for the ten-logarithm for 2.
So, according to the usual theory, this means 100.301 = 2. I'll wager that this is, apart from geometry (natural logarithms come with certain curve shapes) actually an algebraic shortening of another statement.
10 301 = 21000
10 301 =
1 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
21000 =
1 0715086071862673200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
Can't make out whether it's equal length or not? I combined 000 000 000 into "nine zeros" and than three of these into XXVII zeros, then three of those into LXXXI zeros. I then added back last zeros to the part before the abbreviations, and I dissolved parts to bring the number of abbreviations to the same.
10 301 =
10 000 000 000 000 000 000 000 000 000 000
LXXXI zeros LXXXI zeros LXXXI zeros XXVII zeros
21000 =
10 715 086 071 862 673 200 000 000 000 000
LXXXI zeros LXXXI zeros LXXXI zeros XXVII zeros
But the difference is pretty great? over and above the 10 followed by 300 zeros, there is 715 followed by 297 digits (the calculator made most of the digits into zeros).
So, let's check, how different is 100.301 from 2?
100.301 = 1.999 861 869 632 744 1, off by 0.000 1 and some more.
What if we took 0.30103?
100.30103 = 2.000 000 019 968 104 6, off by 0.000 000 1 and some more.
Twice as accurate, in terms of how many digits, but inexact on the other side. No doubt, for
10 30 103 = 2100 000
the excess or deficit would be even more notable than 715 followed by 297 digits. However, that is because you need greater whole numbers in order to give finer fractions as whole number to whole number ratio, the finer the fraction, the more exact the logarithm, the greater the powers of 10 and 2, the greater the discrepancy. I expressed this fact as "the logarithm" (from the arithmetic standpoint) "is the make believe that pretends to an impossible equality between powers of ten and powers of two, the more exact the logarithm, the less the powers are actually close to equality" ... something for which I was taken for a raving fool who had no idea what he was talking about. A mixture of drug addict and intellectual hybris capable of spouting things out which it is a no-brainer to refute, and then being too intellectually arrogant to step down when refuted by people with no brain, or with no evidence of their brain being effectively used for the task.
There is of course, as I more recently found out, a more geometric approach to what a logarithm is, and it is used to get more exact values, and I have no problem with that. Geometry, unlike arithmetic, actually does have Real Numbers, that are actually real. I was answering the proposition that in arithmetic there is no such thing as a stark transition from one to two. Or in other words, that you can have pi apples or sqrt(2) trees.
This is important in the Creation to Evolution debate exactly how?
Well, bear with me, for the relevance is coming. Two of them.
A chromosome may have 10 000 genes. And it may (conceivably) gradually rise in number of genes until it has 20 000 genes. Invent a mechanism that allows a functioning gene to arise, and the 10 000 genes on the chromosome may be 10 001, and then 10 002 and so on. Eventually that may reach 20 000 genes. But they would still all be on that one single chromosome. How is this important?
Well, once upon a time Kent Hovind made a spoof argument about the tobacco plant being more developed than man, because it had more chromosomes. Man only has 46 chromosomes, the tobacco plant has 48.
The Solanaceae species Nicotiana tabacum, an economically important crop plant cultivated worldwide, is an allotetraploid species that appeared about 200,000 years ago as the result of the hybridization of diploid ancestors of Nicotiana sylvestris and Nicotiana tomentosiformis.
In other words, the tobacco plant has appeared before the Flood, and there was tobacco on the Ark. The 200,000 years ago date is just "lava cooled rapidly during the Flood and trapped excess argon" ...
Now, the spoof argument by Hovind reminded me of a real argument in Fr. Bryan Houghton, a non-order priest who was still not totally a diocesan priest, since "incardinated 'in propriam fortunam' " ... in the French translation of his Unwanted Priest, he inserted pieces of tracts, one of them against Evolution. And he mentioned that French scientists had for a long time hushed up the existence of chromosomes, because this poses a barrier to evolution. As mentioned, augmenting the number of genes on a single chromosome may be gradual, but the transition to two chromosomes, if it occurred at all, would be abrupt. There is no such thing as "one and a half chromosomes" for the same reason that there is no such thing as "half a chromosome" ...
There are also not one million intermediates between vocal communications having one level, the full message = one sound, and vocal communications having three levels, full message expressed with composition of morphemes (these being often, but not always, words), and morpheme being expressed in composition of phonemes, and phonemes holding no meaning of their own. There is exactly one possible intermediate, and that is having two levels.
However, there are two ways between the one and the three levels.
1) full message = sound, first divides into full message = many sounds
then this divides into full message = many words, word = many sounds. Or:
2) full message = sound, first divides into full message = many words, but each word = sound
then this divides into full message = many words, word = many sounds.
Note, we have two alternatives. Then again, adding notionality to pragmatism is another item. Did it happen during the first, the intermediate or after reaching the last stage?
We have six alternatives. Placements of three events. And, again, no hint of gradualism being even possible.
Again, the physiological underpinning of the human speach can be reduced to two items: "fully or at least adequately reached" and "not even adequately reached" ... While gradualism is possible, it can for this purpose be ignored. The question now becomes:
Did the apparatus exist before even first division and before notionality?
Or did it arise after first, second or third of the three events above outlined?
Again, the alternatives aren't bafflingly many, you can't say "any scenario we can't even think of is possible" ... and you also cannot pretend I'm crunching "a million gradients" into too few events.
Two scenarios:
Full message divided into sounds. Full message divided into words that remained divided into sounds.
Full message divided into words. Words divided into sounds.
Six scenarios:
Notionality was added to pragmatism. Full message divided into sounds. Full message divided into words that remained divided into sounds.
Full message divided into sounds. Notionality was added to pragmatism. Full message divided into words that remained divided into sounds.
Full message divided into sounds. Full message divided into words that remained divided into sounds. Notionality was added to pragmatism.
Notionality was added to pragmatism. Full message divided into words. Words divided into sounds.
Full message divided into words. Notionality was added to pragmatism. Words divided into sounds.
Full message divided into words. Words divided into sounds. Notionality was added to pragmatism.
Twenty-four scenarios:
Apparatus ready. Notionality was added to pragmatism. Full message divided into sounds. Full message divided into words that remained divided into sounds.
Notionality was added to pragmatism. Apparatus ready. Full message divided into sounds. Full message divided into words that remained divided into sounds.
Notionality was added to pragmatism. Full message divided into sounds. Apparatus ready. Full message divided into words that remained divided into sounds.
Notionality was added to pragmatism. Full message divided into sounds. Full message divided into words that remained divided into sounds. Apparatus ready.
Apparatus ready. Full message divided into sounds. Notionality was added to pragmatism. Full message divided into words that remained divided into sounds.
Full message divided into sounds. Apparatus ready. Notionality was added to pragmatism. Full message divided into words that remained divided into sounds.
Full message divided into sounds. Notionality was added to pragmatism. Apparatus ready. Full message divided into words that remained divided into sounds.
Full message divided into sounds. Notionality was added to pragmatism. Full message divided into words that remained divided into sounds. Apparatus ready.
Apparatus ready. Full message divided into sounds. Full message divided into words that remained divided into sounds. Notionality was added to pragmatism.
Full message divided into sounds. Apparatus ready. Full message divided into words that remained divided into sounds. Notionality was added to pragmatism.
Full message divided into sounds. Full message divided into words that remained divided into sounds. Apparatus ready. Notionality was added to pragmatism.
Full message divided into sounds. Full message divided into words that remained divided into sounds. Notionality was added to pragmatism. Apparatus ready.
Apparatus ready. Notionality was added to pragmatism. Full message divided into words. Words divided into sounds.
Notionality was added to pragmatism. Apparatus ready. Full message divided into words. Words divided into sounds.
Notionality was added to pragmatism. Full message divided into words. Apparatus ready. Words divided into sounds.
Notionality was added to pragmatism. Full message divided into words. Words divided into sounds. Apparatus ready.
Apparatus ready. Full message divided into words. Notionality was added to pragmatism. Words divided into sounds.
Full message divided into words. Apparatus ready. Notionality was added to pragmatism. Words divided into sounds.
Full message divided into words. Notionality was added to pragmatism. Apparatus ready. Words divided into sounds.
Full message divided into words. Notionality was added to pragmatism. Words divided into sounds. Apparatus ready.
Apparatus ready. Full message divided into words. Words divided into sounds. Notionality was added to pragmatism.
Full message divided into words. Apparatus ready. Words divided into sounds. Notionality was added to pragmatism.
Full message divided into words. Words divided into sounds. Apparatus ready. Notionality was added to pragmatism.
Full message divided into words. Words divided into sounds. Notionality was added to pragmatism. Apparatus ready.
The apparatus is not all that useful unless you have notionality and speech. Speech is not possible without the apparatus. (Applies to production, but even more to hearing and learning).
Notionality is not possible before you have three levels. But three levels are not useful without notionality.
Tell me, if you can, which of the twenty-four scenarios isn't destroyed by one of these observations. Because, in evolution, a thing has to be both useful and possible. Thick fur is useful to keep warm. Keeping warm is possible on some levels even before acquiring thick fur. On the other hand, acquiring thick fur first isn't a too bad thing, before the climate change to the cold or the move to a colder clime strikes, it can be a neutral change, and then it becomes useful. This is a change which has happened in kind after kind. And the thickness of fur actually does really allow for several intermediate gradients. It's a geometric question.
Inventing language is a question of arithmetic changes. Which, as mentioned, do not allow for intermediates between the integers.
Hans Georg Lundahl
Paris
Holy Founders of the Servites
12.II.2025
Sanctorum septem Fundatorum Ordinis Servorum beatae Mariae Virginis, Confessorum, quorum depositio respectivis diebus recolitur. Quos autem in vita unus verae fraternitatis spiritus sociavit, et indivisa post obitum veneratio populi prosecuta est, eos Leo Decimus tertius, Pontifex Maximus, una pariter Sanctorum fastis accensuit.
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