*[This post has been updated from its original form to make it better organized, clearer, and more direct and succinct.]*

Bill Walsh is absolutely, completely, 100%, unequivocally, and in all other ways right in his position on the meanings of “times” and “fold” in his recent disagreement that is as much mathematical as it is semantic.

If I start with $100 and end up with $250, did that money grow 2 1/2 times?

A reporter and I are having a good-natured disagreement: He says yes, and I say no.

The confusion between growing, say, 1 1/2 times and 2 1/2 times comes from the fact that some people use *grow* to mean either multiplying or adding, depending on the situation or, I guess, their whim. But *grow* should not mean to multiply. It should only mean to add a number to a starting value. For example, when you say a child grows by an inch, you don’t mean that their previous height was multiplied by 1 inch. The same logic applies when we say something grows by a percentage. For example, say the height of something grew by 1%. That means that 1% of its previous height was *added onto* that previous height, not that its height was multiplied by 1%. Importantly, that percentage is not just an abstract number but refers to a specific, concrete, physical measurement; a raw quantity, say in inches or meters.

The same applies when we convert percentages into *times*. When Bill Walsh and his friend refer to money growing by 1 1/2 or 2 1/2 times, they are really referring to percentages: *percentages of the starting value*. The only way that multiplication comes into a “growing” calculation is when we use a percentage (or “times” or “factor” or “fold”) to calculate how much gets added. When we say something grows by a certain factor of a starting value, what we mean is that we multiply the starting value by that factor and then add that product onto the starting value.

These concepts can be easily seen with a simple linear transformation. Here is the operation as Bill Walsh and I see it:

Let the function “grows

Ntimes” be the linear transformation such that ifxgrowsNtimes, then

x↦x+Nx(xbecomesNtimeslarger thanx).

With this vocabulary, we see that

“$100 grows 1 1/2 times” means $100 ↦ $100 + (1.5)($100) = $250

“$100 grows 2 1/2 times” means $100 ↦ $100 + (2.5)($100) = $350

“100 grows .5 times (grows by half)” means $100 ↦ $100 + (.5)($100) = $150

Alternatively, let’s define a linear transformation according to Bill Walsh’s friend’s use of the English language:

Let the function “grows

Ntimes” be the linear transformation such that ifxgrowsNtimes, then

x↦Nx(xbecomesNtimesx).“$100 grows 2 1/2 times” means $100 ↦ (2.5)($100) = $250

“$100 grows 1 1/2 times” means $100 ↦ (1.5)($100) = $150

“$100 grows .5 times (grows by half)” means $100 ↦ (.5)($100) = $50

In case you aren’t convinced yet, let’s consider the opposite of growing: shrinking. If we use standard English and refer to shrinking as identical to growing except in the opposite direction, then we can easily define an analogous but opposite operation. But first, note that nothing can shrink by more than 100% (1 time); if something shrinks 100%, there is none left.

Here is how I and, presumably, Bill Walsh would define the shrinking transformation:

Let the function “shrinks by a factor of

N” be the linear transformation such that ifxshrinks by a factor ofN, then

x↦x–Nx(xbecomesNtimessmaller thanx).

Note the replacement of “*N* times” from the original formula with “a factor of *N*” in the new formula. This is partly because of how English works and partly because of what I said above about the impossibility of shrinking more than 100%; it can sound awkward to use the plural “times” if the factor is less than 1, and if a value shrinks by 30%, we don’t say it shrank by .3 times but rather that it shrank by a factor of .3, or alternatively by 30%. My “shrinking” transformation thus produces:

“$100 shrinks by a factor of .75 (by 75%)” means $100 ↦ $100 – (.75)($100) = $25

“$100 shrinks by a factor of .25 (by 25%)” means $100 ↦ $100 – (.25)($100) = $75

“$100 shrinks by a factor of .5 (by half)” means $100 ↦ $100 – (.5)($100) = $50

Now let’s define a different “shrinking” transformation analogously to Bill Walsh’s friend’s “growing” transformation:

Let the function “shrinks by a factor of

Nbe the linear transformation such that ifxshrinks by a factor ofN, then

x↦Nx(xbecomesNtimesx)“$100 shrinks by a factor of .75 (by 75%)” means $100 ↦ (.75)($100) = $75

“$100 shrinks by a factor of .25 (by 25%)” means $100 ↦ (.25)($100) = $25

“$100 shrinks by a factor of .5 (by half)” means $100 ↦ (.5)($100) = $50

Not only is that usage of English incoherent, but it requires that growing by half be identical to shrinking by half!

“Now, John, you’re setting up a straw man,” you say. “No one uses *shrink* like that.” Then why do they use *grow* like that?

We can conclude that *grows by* and *shrinks by* are equal but opposite operations and that they each require multiplying the starting value by the growing or shrinking factor and then *adding that product* to the starting value. (For the “shrinking” operation, we can use the phrase “adding that product” to mean adding a negative number, which is the same as subtracting a positive number.)

Now, the word *fold*. As far as the word *fold* goes, I thought its meaning seemed clear to me, but its meaning *as used* seems different, and I would typically avoid using it if I were writing a scientific paper and not just editing others’ papers. In my job as a scientific editor, I think every time I’ve ever seen the word *fold*, it has meant “entailing multiplication by a factor of [the number that comes before it]”. In other words, a 2.5-fold increase always is used to mean “multiplied by 2.5 times”. Therefore, people would say both that $250 is 2.5-fold greater than $100 and that $250 is 2.5-fold (of) $100. That makes no sense to me. Well, no, it does make some kind of sense, but it is inconsistent sense. Well, no, it’s consistent mathematically, because regardless of the construction of the sentence, you just always multiply the original number by the fold factor. But it is inconsistent semantically.

Folds are also frustrating when referring to decreases, but I’ve come to accept formerly non-sensical fold decreases and not care anymore. For example, in the real, physical world, nothing can decrease more than 100%. If some quantity decreases 100%, none of it is left, and there is no such thing as negative matter or energy, so it is non-sensical and meaningless to say something decreased by more than 100%. If something decreases by half, 50% of it is left. If something decreases by two-thirds, one-third of it is left.

Well, if you took *fold* to mean “percent of” or “fraction of”, then the most anything could ever decrease would be 1-fold. If something decreased 0.5-fold, that would be decreasing 50%. Etc. That, however, is not how any biomedical research scientist has ever used *fold* that I’ve seen. They say something “decreased 7-fold”, meaning the final value was 1/7th of the original. If something decreased 150-fold, the final value was 1/150th of the original. That’s stupid, but I guess everyone’s consistent, so now *fold* means “involving a factor or ratio of the original value”.

Bill Walsh has experienced similar frustration with *fold*:

My friendly adversary pointed me to a dictionary that defines the verb

tripleas meaning “to increase three times in size or amount.” And there is the-foldmodel. Atwofold increaseis doubling, athreefold increaseis tripling, and so on. To which I respond: None of the dictionaries on my shelves are that sloppy, and those shelves also hold an otherwise wonderful usage book in which the author is tripped up by-fold, insisting that tripling would be a twofold increase. (It’s a special case,-fold, because “a onefold increase” is not only never used but also impossible. You can fold something in two or three or more, but you can’t fold it in one.)

[I would love to know what wonderful usage book that is. —JTP]

His friendly adversary’s dictionary would, unfortunately, agree with most biomedical scientists on the use of *-fold*: they use a twofold increase to mean multiplying by a factor of 2 (doubling), even though multiplying by and increasing (growing) by ought to mean different things. I suppose you could argue that increasing (growing) by can mean adding to *or* multiplying by according to the whims of the author and following no consistent or pre-defined rule, but that is illogical to me and goes against the meanings of the words “increase” and “grow” as I understand them, especially when the word “by” is added after them. (This is a perfect example of why I am largely prescriptivist: so that meanings can stay as consistent as possible and people separated by time and space (and mathematical ability) will mean the same thing when they use the same words. I will stop being prescriptivist when I gain the ability to understand how people can not care that the same words mean substantially, crucially different things to different people.)

If *shrink* is not the perfect opposite of *grow*, *less than* is not the opposite of *more than*, and *grow (by)* can mean two mathematically distinct things, then, well, I don’t know anything and it’s pointless to write or talk about anything.

I am editing a paper right now that uses one-fold, unfortunately it does not provide the original data that would allow me to check if they really mean 2-fold. Scientists are sloppy with this kind of thing: “A two-fold increase” and “Increased by two-fold” are both used. Most of the time they mean x became 2x, yet either could be interpreted as x +2x. In cases like this (no data to compare), I add a note and let the authors sort it out.

I made a bet today that fold was not the same as times except for the first fold. After some debate and a feeble attempt conjuring mathematical evidence, I have a significant headache. After a hefty internet search which has offered little, I cannot wrap my head around folding is the same as times. In reality all folding of anything is still just the same thing so, really, it is just times 1 and all this means nothing….I wish it was that simple. Beyond that simplistic view, If the term “fold” is used for any multiplier, it would be times the “layers” of the fold, right? If that is the case, I will agree that first actual folding as “two fold” or “doubling” is in fact 2 times 1. Beyond that, the exponential increase will grow with a variable exponent of its own. By the next fold, we would actually speaking of the third fold right? If you fold something once getting 2 layers (times 2), the next fold will give you 4 layers, therefore, 3 fold would mean 4 times, right? The next fold, the 4th, would be 8 layers meaning that “4 fold “is actually “8 times”…….right? At the fifth fold we are at 16 times and you will see where I am going with this, why I have a headache and why I wish I didn’t make the bet in the first place.

I would love to read any argument or agreement on this.

Ben is exactly correct (although we might disagree with what the first fold means, for me I am inclined to think that 1-fold increase is a multiplication by 2). It is ridiculous to use the term “fold” to mean “multiplication by,” although clearly it is the common practice . We need to find the origin of the use of “fold” in reference to increasing quantities, and see if in fact at some point people became lazy and started conflating “fold” and “multiplied by,” or if it always referred to simple multiplication. On the surface to origin of the term “fold” seems obvious to me: you start with a piece of paper (0-fold, multiple-1), and fold it in half (1-fold, multiple-2). Then you fold again in half (2-fold, multiple-4), then again and so on. The relationship between “fold” and the multiple you now have of the original square section of paper is multiple = 2^fold. Every time you “fold” increase, you double the quantity. Does someone else have a better explanation of the origin of the term “fold” in reference to increasing quantities?

Yes, I think that’s exactly where the term “fold” for multiplication comes from, though I don’t think it should be limited to powers of 2. It seems clear enough to me that a “fold” is, or should be, 100%. I don’t have much justification for that other than that’s how it’s used, or at least should be used if everyone were consistent about it. So a 1.5-fold increase would be a 150% increase and a 7-fold increase would be a 700% increase. One good reason to eschew the 2^fold definition is that it’s somewhat rare that we’d be referring only to increases (or decreases) that follow a power-of-2 pattern. I mean, those aren’t

rare, but regular old 150% and 700% etc. increases are much, much more common, so there’s no problem appropriating this term to apply to those increases as well. And I don’t have any evidence, certainly in recent history, that “fold’ was ever restricted to power-of-2 increases.I am translating a scientific text related to BPA which was written by a Polish scientist and I have the same problem. It says “one-fold higher” and I think it means it increased by the same amount of the original. Namely it means, it increased from 100 to 200, imho.