Spuds Illustrated

Image: Wikimedia Commons

I just found this visual explication of the potato paradox — if potatoes are 99 percent water by weight, and you start with 100 pounds of potatoes and let them dehydrate until they’re 98 percent water, what’s their new weight?

The surprising answer is 50 pounds. Blue boxes represent water, orange non-water. So to double the share of the non-water portion we have to halve the amount of water.

(I had thought it was the setting that made this so confusing, but it turns out real potatoes are 80 percent water! So it’s not as outlandish as I’d thought.)

Sideways Music

It’s sometimes contended that time is one of four similar dimensions that make up a single manifold that we call spacetime. The four dimensions are orthogonal to one another, and though humans view one of them, time, as distinct from the others in various ways, it’s not intrinsically different.

Philosopher Ned Markosian offers a novel argument against this view: If aesthetic value is an intrinsic feature of an item, and if the four dimensions of spacetime are indeed similar, then rotating an object shouldn’t change its value. Turning a van Gogh painting 90 degrees doesn’t alter its beauty (though we may now have to turn our heads to appreciate it).

But turning a piece of music “out” of time, so that the notes of its melody, for example, occur all at once, changes the aesthetic value of the piece. “Whereas the original series of events had some considerable positive aesthetic value … the resulting series of events has either no aesthetic value or, more likely, negative aesthetic value. … Hence we have a powerful modus tollens argument against The Spacetime Thesis.”

(Ned Markosian, “Sideways Music,” Analysis 80:1 [January 2020], 51-59; and Sean Enda Power, Philosophy of Time: A Contemporary Introduction, 2021.)


Corresponding with Leibniz about his method of infinite series in 1677, Isaac Newton wanted to advert to his “fluxional method,” the calculus, without actually revealing it. So he used an unusual expedient — after describing his methods of tangents and handling maxima and minima, he added:

The foundations of these operations is evident enough, in fact; but because I cannot proceed with the explanation of it now, I have preferred to conceal it thus: 6accdae13eff7i3l9n4o4qrr4s8t12ux. On this foundation I have also tried to simplify the theories which concern the squaring of curves, and I have arrived at certain general Theorems.

That peculiar string is an inventory of the letters in the phrase that Newton wanted to conceal, Data aequatione quotcunque fluentes quantitates involvente, fluxiones invenire; et vice versa, which means “Given an equation involving any number of fluent quantities to find the fluxions, and vice versa.” So “6a” indicates that the Latin phrase contains six instances of the letter A, “cc” means that there are two Cs, and so on. In this way Newton could register his discovery without actually revealing it — the fact that he could present an accurate letter inventory of the fundamental theorem of the calculus proved that he’d established the theorem by that date. (More details here.)

Robert Hooke had used the same resource in 1660 to establish priority for his eponymous law before he was ready to publish it. And Galileo first published his discovery of the phases of Venus as an anagram. The technique today is known as trusted timestamping.

(Thanks, Andy.)

Cause and Effect

When we are praying about the result, say, of a battle or a medical consultation, the thought will often cross our minds that (if only we knew it) the event is already decided one way or the other. I believe this to be no good reason for ceasing our prayers. The event certainly has been decided — in a sense it was decided ‘before all worlds.’ But one of the things taken into account in deciding it, and therefore one of the things that really causes it to happen, may be this very prayer that we are now offering. Thus, shocking as it may sound, I conclude that we can at noon become part causes of an event occurring at ten a.m.

— C.S. Lewis, Miracles, 1947

Lewis adds, “Some scientists would find this easier than popular thought does.” In his 2016 book Time Machine Tales, physicist Paul J. Nahin writes, “It is a view that does find much support in the block universe interpretation of Minkowskian spacetime. Lewis never mentions the block concept by name, but it is clear that he believed in the idea of God being able to see all of reality at once.” See Asking Back.

Cistercian Numerals

In the 13th century, Cistercian monks worked out a system of numerals in which a single glyph can represent any integer from 1 to 9,999:

Image: Wikimedia Commons

Once you’ve mastered the digits in the top row, you can represent tens by flipping them (second row), hundreds by inverting them (third row), and thousands by doing both (fourth row). And now you can combine these symbols to produce any number under 10,000:

Image: Wikimedia Commons

The monks eventually dropped the system in favor of Arabic numerals, which reached northwestern Europe at about the same time, but it was being used informally elsewhere as recently as the early 20th century.


The ancient Chinese philosopher Gongsun Long appeared to claim that a white horse is not a horse:

Is ‘a white horse is not horse’ assertible?

Advocate: It is.

Objector: How?

Advocate: ‘Horse’ is that by means of which one names the shape. ‘White’ is that by means of which one names the color. What names the color is not what names the shape. Hence, one may say ‘white horse is not horse.’

Objector: If there are white horses, one cannot say that there are no horses. If one cannot say that there are no horses, doesn’t that mean that there are horses? For there to be white horses is for there to be horses. How could it be that the white ones are not horses?

Advocate: If one wants horses, that extends to yellow or black horses. But if one wants white horses, that does not extend to yellow or black horses. Suppose that white horses were horses. Then what one wants [in the two cases] would be the same. If what one wants were the same, then ‘white’ would not differ from ‘horse.’ If what one wants does not differ, then how is it that yellow or black horses are acceptable in one case and unacceptable in the other case? It is clear that acceptable and unacceptable are mutually contrary. Hence, yellow and black horses are the same, one can respond that there are horses, but one cannot respond that there are white horses. Thus, it is evident that white horses are not horses.

Interpretations vary; one explanation is that the conundrum blurs the distinction between identity and class, exploiting an ambiguity in the Chinese language — certainly the expressions “white horse” and “horse” do not have identical meanings, but one can refer to a subset of the other.

Whether the philosopher was serious isn’t clear. His other paradoxes include “When no thing is not the pointed-out, to point out is not to point out” and “There is no 1 in 2.”

More trouble with horse color.

03/08/2024 UPDATE: A Swedish Facebook meme of 2012: Horses are a fruit that does not exist. (Thanks, Mikael.)