With all due respected to the illustrious Dr. Iverson, he was in loony-land with his two versions of minus.
The notation "-3" does not mean "negative three", which, as my freshman math professor, a lover of English, pointed out, is a solecism, since "negative" is an adjective, and three is and always will be a positive number. There is no negative version of three, there is simply three, a positive number, and its opposite, a negative number, which is firmly not three, and not a species or kind of three. The name for the opposite of three in English, properly, is of course, "minus three" - simply the reading of the signs by which we indicate the number. Now for three and its opposite, this is not a terribly important point. But my professor would insist on it because with *other *notations, it *does * matter. For when we speak of the square root of four, we are naming two different numbers at once ambiguously, and we can speak of the positive square root of four (two) and the negative square root of four (minus two). For the negative square root of four is *not *"the opposite of the square root of four", which is just as ambiguous as saying "the square root of four". (The opposite of the negative square root of four is, of course, the firmly positive two.) The systematic ambiguity which Iverson was on to is not about numbers like three at all; there is no sense in which the minus sign in "-3" is any different from the one in "-*x*". The former is not "part of the numeral"; it's a way of indicating the opposite of three, just as -*x* indicates the opposite of *x*. But when we need to distinguish the two meanings of √4, we do find signs useful, and we can write -√4 and +√4 in order to clarify whether we mean the positive or the negative square root of four. And this indeed can matter when the expression under the radical is more complex. It really matters when we are concerned with √-1, for which the two values are neither positive nor negative, and so we cannot properly say +√-1 or -√-1, even though we certainly can say -*i *and *i. *(And the key then is the systematic symmetry of the complex numbers, in which any theorem that can be proved about the complex numbers can be proved for their conjugates as well, which is certainly not true for 2 and -2!) Now, whether minus three is a single idea or two, I submit that counting ideas is a thing that leads only to madness. "As I was going up the stair, I met a man who wasn't there. He wasn't there again today; I wish, I wish, he'd go away!" When there is a book on the table, is there one thing or many? What is a "composite idea" and which ideas are "simple"? (And how our ancestors would be horrified to hear someone say that minus three is a simple idea!) So it would be nice to have one symbol in the case of -√4, and a different one for both -2 and -*x. *For the case of *i *and -*i *the case is even weirder, for our theorems don't depend on the difference at all, as long as we have a difference. Now if you're still reading, and you say, "but Thomas, -2 is negative, but for -*x* it might be or it might not be!" Yes, certainly, but this is because two is positive and *x *is unspecified. The difference is the difference between "2" and "*x", *between a determinate quantity and an indeterminate one, and not anything to do with what "-" means: in both cases it means "the opposite of". You might think that in the case of -2 the sign is asserting that the number is negative, and that this is a useful thing. Well, this is exactly what is going on with -√4: but not in the way of asserting that √4 is negative, but rather of *choosing *from two values the one which is negative. -2 is doing nothing of the kind; it is not as though 2 indicated two different things ambiguously. Yours, pedantically. Thomas On Fri, Aug 19, 2016 at 4:00 PM Michael Jones <michael.jo...@gmail.com> wrote: > My answer is philosophical, and possibly not what is wanted, but it is > informed. Here goes: > > > > Backstory: Unary Minus, as taught in computer science and implemented in > languages and instruction sets and so forth is slightly confusing and often > confusingly described. > > > > x := -y // means get the value of y, negate that, and assign to the > location known as x. > > > > This is what we commonly think of when seeing unary minus. It is proper. > However this… > > > > x := -3 // does NOT mean get the value 3, negate that, and assign to the > location known as x. > > > > …instead it means “assign the value ⁻3 to x”, where negative 3 is a single > idea, not two as in (Negate 3). The raised minus sign is from Kenneth > Iverson, who used it in APL to make the point that the ‘⁻’ and ‘-‘ have > entirely different meanings. > > > > There is a Wikipedia article about this but its scholarship is, well, more > about usage than history. They talk about the raised minus as an elementary > school teaching tool (which it may be) but they leave it there with little > suggestion of its illustrious past. Yet, to their credit there is a handy > example: > > > > ⁺3 − ⁻5 becomes ⁺3 + ⁺5 = ⁺8 > > > > where you might imagine a teacher saying “positive three minus negative > five becomes…” which is nice. (A sophisticated college CS teacher or parser > coder might instead say “plus 3 minus minus 5 becomes…” ;-) We don’t have > the raised minus to say “here is a negative number” so we make use of the > unary rather than binary minus to suggest negative. You use what you have. > > > > Now, on to the original question: > > > > Standard floating point output since the 1950’s has allowed printing of > either blank or “+” as the prefix for positive floating point values and, > necessarily, the lowered minus “-“ for negative values; we get the ‘+’ with > a printf format like “%+10.4f”. > > > > It is handy to be able to parse what is output, such as numerical > approximations and weights in formula source code, so the parsing of > “3.1415” was made to accept “+3.1415” for the same meaning, and of course > “-3.1415” since we lack the raised minus. > > > > Since floating point values accept the stylistic ‘+’ prefix, so do integer > values. > > > > ..and here we are today, some 50 years later. > > > > P.S. If you visited Thomas Payne’s bookshop in London (in the 1750s) you > could have bought your own copy of Francis Maseres’ *“A dissertation on > the use of the negative sign in algebra: containing a demonstration of the > rules usually given concerning it; and shewing how quadratic and cubic > equations may be explained, without the consideration of negative roots. To > which is added, as an appendix, Mr. Machin's quadrature of the circle.”* > You could have enjoyed much discussion of the new style of writing and its > advantages. > > > > > > *From: *"'Thomas Bushnell, BSG' via golang-nuts" < > golang-nuts@googlegroups.com> > *Reply-To: *"Thomas Bushnell, BSG" <tbushn...@google.com> > *Date: *Friday, August 19, 2016 at 1:36 PM > *To: *Axel Wagner <axel.wagner...@googlemail.com>, Rob Thornton < > rthornton...@gmail.com> > *Cc: *golang-nuts <golang-nuts@googlegroups.com> > *Subject: *Re: [go-nuts] Unary + > > > > In C, the type is promoted. But this ain't C. No clue what the point is in > Go. It would function as a compile-time type-check for "numeric" I guess, > but what would be the point? > > > > > > On Fri, Aug 19, 2016 at 1:32 PM 'Axel Wagner' via golang-nuts < > golang-nuts@googlegroups.com> wrote: > > x would already be signed during implicit type inference. The inferred > type is either the one specified in an argument or declaration, or it's the > default type (which is int, thus signed). > > > > I don't know the answer to your question though. I didn't even know this > existed… > > > > On Fri, Aug 19, 2016 at 9:54 PM, Rob Thornton <rthornton...@gmail.com> > wrote: > > What is the purpose of the unary '+' operator? In both C and Go they are > syntactically correct but neither generate instructions to modify the > expression. > > The '-' operator obviously generates a neg or sub from 0 instruction to > negate the result by inverting the expression with an add with carry. > > I know of no single instruction or pair of instructions to reliably ensure > any value or expression is always positive so why include this unary > operator at all? Is there a purpose I'm missing? Or is it just for clarity > or force an implicit variable to be signed? > > As in: > > x := +5 > > Would this ensure that x must be signed during impicit type inference? > > If I read the specification correctly, this would also affect the > resolution of untyped constants. Is this correct? > > -- > You received this message because you are subscribed to the Google Groups > "golang-nuts" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to golang-nuts+unsubscr...@googlegroups.com. > For more options, visit https://groups.google.com/d/optout. > > > > -- > You received this message because you are subscribed to the Google Groups > "golang-nuts" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to golang-nuts+unsubscr...@googlegroups.com. > For more options, visit https://groups.google.com/d/optout. > > -- > You received this message because you are subscribed to the Google Groups > "golang-nuts" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to golang-nuts+unsubscr...@googlegroups.com. > For more options, visit https://groups.google.com/d/optout. > > -- You received this message because you are subscribed to the Google Groups "golang-nuts" group. 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