23 may 2025

23 may 2025

John Bjarne Grover

Aristotle's intuition on natural logarithms

On 24 april 2025, I wrote in the 'Brief comments':

In my article 'The submorphemic Aristotle' there is a discussion under αἴτια = the plus/minus sign - under Example 1 – Physics 207b-35 - the comment to the example, wherein I write "there is the special idea that the logarithms are normally between 0 and 1". It must mean that this is the logarithms to the probability values (normally) between 0 and 1. It is a strange and interesting phenomenon that the more one reads Aristotle on these matters and tries to understand it, the more difficult it is to understand what a logarithm really is. What was easy in school suddenly becomes so difficult. That could be because Aristotle probably did not have the mathematical tools available for understanding the logarithms which I take his ὕλη to be about. Could be there was a gap there which perhaps Euclid later tried to fill in.

This excerpt about Aristotle's concept of 'aitia' is this:

Example 1 – Physics 207b-35

Text:
ἐπεὶ δὲ τὰ αἴτια διῄρηται τετραχῶς, φανερὸν ὅτι ὡς ὕλη τὸ ἄπειρον αἴτιόν ἐστι, καὶ ὅτι [208a] τὸ μὲν εἶναι αὐτῷ στέρησις, τὸ δὲ καθ' αὑτὸ ὑποκείμενον τὸ συνεχὲς καὶ αἰσθητόν.

Translation:
Since the plus/minus signs are divided into four parts, it appears that just as the logarithm [ὕλη] is the 'ἄπειρον' (countless, infinite) plus/minus sign, and that on the one hand being (existence) in itself is a privation, on the other hand according to the denominator [ὑποκείμενον] it is continuous and perceptible.

Comment:
This tells in fact precisely this: The general idea is that the human reality is in itself a privation with important parts negated, on the other hand there is the special idea that the logarithms are normally between 0 and 1. There are four summation signs in the formula which means also four summations to one or above/below - which is why the plus/minus sign is a variable.

My comment tells "the special idea that the logarithms are normally between 0 and 1". If these logarithms (Aristotle's ὕλη) apply to the probabilities of the distribution, then it makes sense to say that "the logarithms are to the probabilities normally between 0 and 1", otherwise is sounds strange. But is it really? I hesitated as to whether I should correct this apparent blunder but have let it be left as it is. The reason is that if it should be correct, then it means that the sum of probabilities normally will be between 1 and the base of the logarithm (what I take Aristotle's μορφή = 'form' to be about). Then the interesting question will be if there are traces of this in other parts of Aristotle - I bring the following from the discussion of the στοιχεῖον -

Example 7 – Physics 189b-17

Text:
τὸ μὲν οὖν τρία φάσκειν τὰ στοιχεῖα εἶναι ἔκ τε τούτων καὶ ἐκ τοιούτων ἄλλων ἐπισκοποῦσι δόξειεν ἂν ἔχειν τινὰ λόγον, ὥσπερ εἴπομεν, τὸ δὲ πλείω τριῶν οὐκέτι• πρὸς μὲν γὰρ τὸ πάσχειν ἱκανὸν τὸ ἕν, εἰ δὲ τεττάρων ὄντων δύο ἔσονται ἐναντιώσεις, δεήσει χωρὶς ἑκατέρᾳ ὑπάρχειν ἑτέραν τινὰ μεταξὺ φύσιν• εἰ δ' ἐξ ἀλλήλων δύνανται γεννᾶν δύο οὖσαι, περίεργος ἂν ἡ ἑτέρα τῶν ἐναντιώσεων εἴη.

Translation:
If the summations [τὰ στοιχεῖα] are said/thought to be three, they will be expected/considered (?) to be from these and from such others, if they have any logos, such as we distribute [εἴπομεν], but more than three they are not: for, on the one hand, it is enough with one for being acted upon [πάσχειν], and if there were four they would be in two contraries (couples), they would be attached inbetween mutual subscript/indexation each to the other during the 'expansion'; if, on the other hand, they could generate each other being two, the other one of these contradictions would be 'over-careful' (= 'overdetermined'?).

Comment:
It could mean that the fourth element is the fire = 'heat' that works on it for drying it only after it has been smeared on the glass. One could think of air and fire as being less essential. Aristotle's account of the two pairs of στοιχεῖα makes sense for the summation signs in the formula.

Excerpts like these can be telling - here it would mean that Aristotle seems to suggest that the sum of probabilities can go up to 2 or maybe 3 but not further. If so, that should mean that Aristotle has an intuition of natural logarithms with base 2,718... If that should be the case, then my comment to his Physics 207b-35 is right. This is the reason why I left it as it was - since that would be a very interesting philosophical point. Aristotle of course had no concept of logarithms but since his philosophy on substances and what I take to be summation of probabilities is strongly affected by a phenomenon of new substances created in the course of the count, it follows that natural logarithms would be the natural format for Aristotle's logical intuition. I would say that this could be very interesting philosophically.

Logarithms can have any base. For example, if the base is 10, then the logarithm to the number 100 is 2 - because 10² = 10 in the power of 2 is 10*10 = 100. What is the number 100 expressed in the base 2? That is 6.644 - which means that which is 2^6,644 = 2 in the power of 6,644 gives 100. If the base if e = 2,71828... then the logarithm is 4,605, which means that 2,71828^4,605 = 100. This is often written ln(100) = 4,605.

Then what is so special about the 'natural logarithm'? It is the special properties of the base called the constant 'e' = 'Euler's number' which is said to be an irrational and transcendent number with an infinite number of decimals - it wedges inbetween all numbers one can think of in that area and is so to speak in touch with 'another reality'. If one has a curve in the XY coordinate system, one can draw a tangent to this curve at any point. Natural intuition tells that there can be one and only one tangent to such a curve at any point. This tangent will have an angle of inclination which is telling of the 'slope' of the curve in just that point. This angle of inclination can be given a number which is the ratio between the two right sides of a triangle (in an xy-coordinate system) where the tangent is the hypothenuse: If the horizontal side (x) of this is bigger than the vertical side (y), then the ratio x/y is smaller than 1 (between 1 and 0), otherwise it is bigger. If the hypothenuse is the tangent to the curve, then the 'slope' of the curve in a single point will be just this x/y which is itself a number which can be called a 'y' to the 'x' in that point, and this 'y' will go up and down as the curve changes with the 'x', and these changes in the 'y' with the 'x' will be a new curve. It is the relation between the two curves that is the theme of differential calculus: The second curve will be a derivative of the first, and the first will be an integral (antiderivative) of the second. It is a peculiar fact that the first curve qua integral to the first will be the same as the value of the area under the second - between the curve and the x-axis.

This means that a derivation of a curve in principle will be the same as a reduction in dimensions - from a two-dimensional area to a one-dimensional line (however curving). This is really also what a logarithm is - since 'a' multiplied with 'b' - that is a*b - can be converted into log(a)+log(b) - that is, logarithms reduce the task of multiplication to a matter of addition. (Addition is learnt first in school, multiplication thereafter). A multiplication is the same as an area and an addition is the same as a line.

What is so special about the natural logarithm - that is, with the base 'e' = 2,71828... - is that it is its own derivative.

This means that if an 'ex nihilo' object is believed to be something that 'falls down' into the human reality from a higher dimensionality, the natural logarithm (with base 'e') will be precisely the format for an intuitive understanding of what can be the mathematical nature of such an 'ex nihilo' object.

More specifically, the base 'e' for the natural logarithm will be that special number which makes for just that property that the function y = e^x is its own derivative. There is only one number which has this property - that is Euler's number.

Not surprisingly, therefore, since objects so to speak can fall down from higher to lower dimensions through this number as the logarithm-base, it is assumed that the number is 'transcendent' and 'irrational'.

Natural logarithms (on the base 'e') are precisely that single thing - the function which is its own derivative. If a math exposition could try and avoid that simple explanation (as if wriggling around in jargon avoiding this single fact), it could be because the simple fact reduces the whole complex with at least one dimension untill it falls down on that single little fact. Could be also that single little fact will tend to bring 'ex nihilo' formation of matter closer to the reality, and humans often try to avoid that.

Aristotle clearly could have conceived something close to natural logarithms on the base 'e' by sheer natural intuition when he was making philosophy on the concept of 'different substances'. This could therefore explain some of his considerations - such as e.g. the above-mentioned from Physics 189b-17 - which in fact could be seen to concern some intuition on formalities related to derivation and integration for a conclusion 'close to 3' - could be even 2,71828...

A poem of mine from the first half of the 1990's

I was taking off my wristwatch
while preparing for the night
and I put it in my pocket
as I quenched the burning light.

As I pulled my trousers off
the watch fell down in front of me
as a fish escapes the bucket
jumping back into the sea.

It lay there on the carpet
and its silvery watchband shone
in the darkness like some angel
who's afraid to be alone.

As I slipped under my blanket
I could see the evening star
shine with pallor through the curtain
and the window set ajar.

In the haven, on a liner,
some young man sings as he boards
and I know under my blanket: Clock
is time of broken swords.

I think I had this on this homepage of mine some years ago but now I cannot find it anywhere there.

My 1994-95 novel 'The Dreamer' is possibly the world's most plagiarized novel. If it should have been published om paper as a separate book, it could have had Eva Xiuxia Jing's artwork (source) on the cover.

Alternatives: 1) who's afraid of being alone. 2) onto a liner.

* * *

Some days ago I found my PO Box in Szolnok unlocked - I dont think I had left it unlocked - probably I had not.

Some time ago I think I read that current hungarian PM Viktor Orbán (or was it his party) considers Péter Magyar (means 'Peter Hungarian') an opposition politician who tries to lead Hungary into an orange revolution like the one they had in Ukraine and elsewhere. I notice this 'M.Ajar' which perhaps could be read from my poem. In collaboration with Microsoft Windows?

I had written the beginning of this little comment down to "I would say that this could be very interesting philosophically" - and added today the rest from "Logarithms can have any base". The first part was written and is stored on my computer in a file saved on 21 may 2025 at 03:26 AM.

Today 23 may there are news of a recent earthquake outside Hokkaido in Japan - it was yesterday afternoon 2025-05-22 at 21:28:01.8 UTC. It was 23 km off Urakawa which is a name written with the characters 浦河町 which can be interpreted 'Pu-he-ting' = 'riverbank raised-path-between-field'. (The 'Puheting' must not be confused with the norwegian parliament called 'Storting' = 'Bigthing'). The name sounds like a paradox of a conversion of the downsloping banks of a river plus the upsloping sides of a raised path between fields - that is, if it be about a paradox called water vs earth. Norwegian 'or[d]-a-kava/kaffe' can also be called 'spå i grut' - for divinations perhaps akin to the use of the oracle bone script in ancient China - see my study of some oracle bone script signs.

For 'spå i grut', it is true that I also mumbled for myself yesterday around perhaps 2-3 o'clock (or thereabout) in the afternoon in the afternoon while still on a riverbank after having crossed the bridge the title 'Poikilothron' of the best preserved poem of Sappho. One should not normally expect to find that there could be an automatic surveillance and 'decipherment' of my mumble for turning it into earthquake. A 'sap-foe'.

'Vi river i' is a norwegian expression telling 'we will pay for it' - it does not mean 'we go into the river' (such as the aircrash into the Potomac river in Washington DC some time ago). 'River' is a word meaning 'tearing ['apart'/'into']' - but it can also be used ('river i') about paying for somebody else. (I notice also the comment of 30 april 2025 about 'adobe'. My house in Szolnok is made of adobe).

This comment was originally published in two parts: The one on Aristotle on 23 may 2025 and my poem of high relevance thereto on the day after 24 may 2025 - but both were in fact published late on 23 may 2025. Today 24 may I put the two comments together in one of 23 may and made this new comment of 24 may 2025 the comment of this day.