# Why are distributions "tempered"?

Google N-Gram shows that both "tempered distribution" and "temperate distribution" are used in English, but the first version significantly prevails, and usage of the second term declines.

Schwartz himself seems to have used this term for the first time in his paper of 1951, Analyse et synthèse harmoniques dans les espaces de distributions, Canad. J. Math. 3 (1951), 503–512, doi:10.4153/CJM-1951-051-5, and the French term was "distributions tempérées".

I am not a native English speaker, but I understand "temperate" as a synonym of "moderate", which makes sense to me as a name of those distributions. For example, we say "temperate climate", not "tempered climate". While "tempered" seems to be related to metallurgy, namely to a process making steel hard. What does tempering of steel have to do with distributions?

Why does the name "tempered" win in English?

• I don't know the etymology of the mathematical adjective, but I have understood the name in the sense of "to soften or moderate", as in the 1st definition of the verb: merriam-webster.com/dictionary/temper
– Neal
Jul 19, 2021 at 15:09
• The well tempered clavier. Tempered sounds better in both cases to my ear; and in this case also one could imagine temperate as closer to "temperierte". Jul 19, 2021 at 15:38
• Temperate feels more like a property of an entity with its own agency that happens to take on a moderate character. Tempered feels more like the property of an entity that is under outside control and has been influenced to take on a moderate character. Hence, the weather is temperate, but the steel, the clavier and the distribution are tempered. Jul 19, 2021 at 16:20
• (Native U.S. English speaker...) I'd agree with @MichaelEngelhardt, thinking of both "tempered" and "temperate" as just adjectival forms of the verb "to temper", after all. Jul 19, 2021 at 16:27
• Mathematical terminology in English is full of slight mistranslations from French (and German). Jul 19, 2021 at 17:39

Can someone explain, why in English the name "tempered" wins?

Presumably because that’s how the inventor himself translated it (French past participle to English past participle), on e.g. p. 188 of Schwartz, Laurent, Mathematics for the physical sciences, Collection enseign. des sciences. ADIWES Internation Series in Mathematics. Paris: Hermann & Cie.; Reading, Mass. etc.: Addison-Wesley Publishing Company. 357 p. (1966). ZBL0151.34001:

A distribution $$\mathrm T$$ (that is to say a continuous linear form on $$\mathscr D$$) is termed a tempered distribution if it may be extended to a continuous linear form on $$\mathscr S$$.

The usage was apparently already well-established by 1956 when G. Mackey reviewed a French paper of F. Bruhat and wrote (first occurrence of the term in MathSciNet):

The representations concerned are assumed to yield “tempered representations” when restricted to the Abelian normal subgroup being studied. Here tempered means being not too badly unbounded in a precise sense suggested by Schwartz’s definition of tempered distribution.

• Well, in fact I already checked sites.mathdoc.fr/OCLS and it seems the first piece of writing by Schwartz in English about his theory is "Applications of distributions to the study of elementary particles in relativistic quantum mechanics" in 1960. On google books the relevant pages are not visible, but by doing a search within the document, one can see that he indeed used "tempered" and not "temperate". That being said, I trust Laurent Schwartz with his math but not so much with his English. I don't think one should follow his example about this choice of terminology. Jul 19, 2021 at 18:23
• Interestingly I. E. Segal uses temperate in his review of L. Schwartz’s 1952 Transformation de Laplace des distributions. Jul 19, 2021 at 19:04

"Tempered" is used not just with steel but with thoughts and emotions:

• "What heaven can be more real than to retain the spirit-world of childhood, tempered and balanced by knowledge and common-sense." (Beatrix Potter, 1896)

• "Nationalism is power-hunger tempered by self-deception." (George Orwell, 1945)

• "Regret for the things we did can be tempered by time; it is regret for the things we did not do that is inconsolable." (Sydney Harris, 1951)

Perhaps that usage is what inspired the original mathematical users.

• Don’t forget the well-tempered clavier and tempered in the sense of ‚tuned‘ Jul 20, 2021 at 12:39

I think Michael, in the comment above, gave a convincing explanation for when to use "temperate" vs. "tempered". The answer by Matt also consolidates this point of "tempered" being associated to the result of an external action, since in all three examples the word "tempered" is followed by "by". Now where I disagree is about the conclusion to be drawn from this.

The delta function $$\delta(x)$$ is temperate because that's what it is. It is not like it originally existed in the form $$\delta(x)+e^x$$, and then some deus ex machina came and tampered with it (sorry couldn't resist) by subtracting the exponential, and finally made it into a tempered distribution.

• I agree it's not completely clear-cut for the distribution. However, is a distribution something that exists independent of a mathematician constructing it? Does it have any agency in determining its properties? Probably, to really get into the weeds, one should also distinguish between the concept "tempered distribution" and any particular instance of such. Jul 19, 2021 at 17:59
• Ah...I guess this would lead to classic debate plato.stanford.edu/entries/platonism-mathematics I think I'll pass on that one. Jul 19, 2021 at 18:07
• Yes, I don't think that discussion would lead to a satisfying conclusion. Jul 19, 2021 at 19:56
• Perhaps an analogous case: we speak of "tame" ramifications, not "tamed" ramifications. Jul 20, 2021 at 3:27
• Aesthetically, I'm not averse to thinking of tempered distributions metallurgically: molten ore (a smooth test function) is left to cool (tend in a limit) and while it is annealing it is tempered (by Schwartz seminorms), with the result a tempered metal blade (tempered distribution), (locally) sharp, (globally) rigid, but not brittle (allowed to grow moderately). Jul 20, 2021 at 7:40

I've always understood tempering as a process to enable something to be more easily molded into something useful--to make it more malleable and robust--in this case the Fourier transform. As nLab puts it: The main property is that the Fourier transform of a TD is well-defined, and is itself a TD; and that it naturally extends the standard FT. This makes TDs the natural setting for solving (linear) PDEs.

Consistently, in metallurgy, tempering increases the ductility of a material, i.e., "the degree to which a material can sustain plastic deformation under tensile stress before failure," making it more useful.

Similarly, to temper one's emotions is to guide them, shape them, mold them, into productive channels, or at least less destructive / disruptive ones.

'To temper' has a long history. From Oxford Languages:

In Latin, temparare--to restrain, moderate $$\to$$ Old English, temprian, and Old French, temprer--to temper, moderate.

Old English temprian ‘bring something into the required condition by mixing it with something else’, from Latin temperare ‘mingle, restrain’. Sense development was probably influenced by Old French temprer ‘to temper, moderate’. The noun originally denoted a proportionate mixture of elements or qualities, also the combination of the four bodily humors, believed in medieval times to be the basis of temperament, hence temper (sense 1 of the noun) (late Middle English).

(I'm pretty sure Schwarz understood French.)

We use, in modern English, 'a temperate climate' to mean a moderate climate between tropical and harsh northern climates--a comfortable mixture of the two. We can say, "The climate of the coastal regions is moderated by the cool waters of the Pacific," but never, "The climate of ... is temperated by ...," rather, "The climate of ... is tempered by ... ." Temperate is purely an adjective like 'mild' whereas tempered is a verb (past participle) used as an adjective meaning to have been tempered--same grammatically as burned in 'a burned / burnt car'. Even the pronunciation of moderate changes according to whether it is being used as an adjective or as a verb or the past participle adjective moderated.

(Maybe a native French speaker can comment on parallels in the grammar in French.)

From all the considerations above, I see the delta 'function' as an construct of Heaviside and Dirac that has been molded into one amenable to the tastes of purist mathematicians (It's NOT a function!--the shrill mantra)--a morphing not really necessary for pragmatic physicists and engineers--by Schwarz and his theory of distributions. In that sense the delta function as a distribution that has been tempered by Schwarz and his theory seems fitting, just as the climate of coastal southern California is tempered or moderated by the Pacific waters. Of course one is free to say the climate along the coast is temperate, but that doesn't stress an agent and action that results in that quality.

To end with the Bard:

1591, William Shakespeare, Henry VI, Part 1, Act II, Scene 4, [22]:

Between two blades, which bears the better temper: […] I have perhaps some shallow spirit of judgement; But in these nice sharp quillets of the law, Good faith, I am no wiser than a daw.

• Did anyone suggest to use the non-word “temperated”? The question is about “tempered” vs. “temperate”, both of which undeniably exist in English, and their usage in the given context is syntactically correct. Jul 27, 2021 at 10:28
• The answer illustrates the intrinsic differences between temperate and tempered by examples that a literate native English speaker should already know but not necessarily a nonnative speaker. In understanding that difference it's important to stress the differences in form and usage including forms that are never used. The OP is mistaken, e.g., in that a tempered/moderated climate would be acceptable in the sense of a climate tempered/moderated by the Pacific waters. This addresses comments on the differences between temperate and tempered, Jul 27, 2021 at 10:52
• @EmilJeřábek, when you teach logic, do you only talk about 'and' and 'or' never 'nand' or 'nor'? I'm also free to go beyond anyone's suggestions. Jul 27, 2021 at 17:01

For me it's psychological: these are the distributions that are (well-)behaved, or (well-)tempered as opposed to the ill-tempered ones; "temperate" wouldn't work in "well-temperate" or "ill-temperate".