Good answer! Allow me to stand you a shot of high proof whisky.
Good one. But that is an example of the older meaning of proof. A good example. Cf. photographic proof, nothing like a mathematical proof.
I find it bizarre the meaning of “proof” as commonly used by judges in court. At preliminary hearings, for example, we often read that judges will decide to dismiss a case, or not, according to whether the plaintiff has “proved” he has a case.
In other words, you have to “prove” your case to the judge first, in some sense, after which you might get a chance to “prove” the same to the jury.
In criminal cases, a preliminary hearing takes the place of bringing an indictment. It’s a proceeding used to demonstrate that the prosecutors have at least sufficient evidence to bring formal felony charges against a defendant. It’s not a trial, and beyond being present, a defendant doesn’t have to do anything.
If there isn’t enough evidence to bring felony charges, then the case is dealt with as a misdemeanor or dismissed before the whole banana of a trial is even contemplated.
IANAL, and also this is a thread hijack, so if you want to continue this aspect of the discussion we should probably move to another thread.
Okay, I suppose I’m forced to go into the niggly details.
It all depends whether you’re talking about classical Latin, medieval Latin, legal Latin, church Latin, or 17th century scientific Latin. The meaning of Latin words can be different from what you expect in different times, places, and contexts.
The legal principle – much misunderstood and misused – is ‘Exceptio probat regulam in casibus non exceptis’ – ‘An exception proves [that there is] a rule in cases not excepted’.
It means that if an exception exists, that is legal proof that a rule also exists for cases not included in the exception.
e.g. If a sign says “No parking 8am-5pm on weekdays” then a court is entitled to assume that parking is allowed at all other times. The implied rule is that parking is allowed, and the sign indicates the exceptions. (You might think that nothing is implied either way about the times not mentioned on the sign, but you’d be wrong in a court of law.)
‘Probat’ here means ‘provides legal proof’. It does not mean ‘tests’.
Newton in his Principia uses probare for mathematical and scientific proof:
Propterea triangula abc, ABC, quæ modo similia esse probavimus, sunt etiam æqualia.
Therefore the triangles abc, ABC, which we have proved to be of similar kind, are also equal.
Et simili argumento probabitur esse KD ad SD in eadem ratione.
And by a similar argument it will be proved that KD ad SD are in the same ratio.
Planetas omnes in se mutuò graves esse jam ante probavimus.
We have already proved above that all the planets are mutually in each other’s gravity.
Yes, but that’s true of pretty much every major discovery in math and science. Without Maxwell, for example, there are no Einstein’s Theories of Relativity.
It would obviously be false to claim that Newton and/or Leibniz invented calculus completely out of thin air, and the reason they both effectively reached the same destination separately is because of all the previous work that was out there in the mathematical zeitgeist. But Newton was the first one to crystallize it all into writing in one place (relatively speaking), following shortly thereafter by Leibniz. That’s how it goes, and has always gone.
“If I have seen further it is by standing on the shoulders of giants.”
And there’s a reason that there are often controversies over who exactly invented or discovered something first. It’s not a coincidence that different parties came to the same conclusions around the same time, it’s because the knowledge and technology of society improved to the point that those things could be discovered, and then it’s just a matter of who puts the pieces together first.
The shoulders-of-giants metaphor can be traced to the French philosopher Bernard of Chartres, who said that we are like dwarfs on the shoulders of giants, so that we can see more than they, and things at a greater distance, not by virtue of any sharpness of sight on our part, or any physical distinction, but because we are carried high and raised up by their giant size.
And lots of others said something like it before Newton.
“If I have made great quotes, it’s because I’m paraphrasing giants.”
So what? I don’t think Newton or anyone else claimed that he originated that idea.
While my wife is a physics geek, I’m more interested in the individuals in the historic sense, even so, I’m not dedicated enough to give a fair evaluation of the assertations.
But I do know the basics of the board (small triumph), so I’ll lend you the skills of my shallow knowledge!
@Hop_David, you can, bar use of various tools to ignore and mute threads and posters, notify a poster of your interest by using the ‘at’ sign as I did in conjunction with your user name. The rest is up to time and the fates (and the interest of the other poster).
@neiltyson, if you have a chance, Hop_David would like to discuss your video and the assertations within.
And regardless of result, I think both of you have made good points in the thread, and look forward to learning more.
[ Aside: Regarding my prejudices, I’m in the Leibniz camp, although fully acknowledge that none of these techniques and uses came out of a vacuum. But I’m in the “if you don’t publish/share your work to the world, I reserve the right to be skeptical of your claim” camp. ]
It’s standard markdown.
This intro may be useful:
It’s also worth while to go through all the options on your profile and set them the way you want.
Whoosh. It was a joke based on the two posts before yours.
As usual, despite all the math classes I took, I’m completely lost in threads like this. But at least I got to geek out over NDT commenting on our board! C’mon, the rest of you are pretending you’re not sitting there with jaws hanging over this, right?
It is undignified to geek out too much, even as awesome as NDT is.
(I mean, I named one of my Star Trek Online characters after him, I’m absolutely a fan.)
Citing Cecil just for fun
Cicero was defending one Bilbo. (No relation to Frodo.) Bilbo was a non-Roman who was accused of having been illegally granted Roman citizenship. The prosecutor argued that treaties with some non-Roman peoples explicitly prohibited them from becoming Roman citizens. The treaty with Bilbo’s homeboys had no such clause, but the prosecutor suggested one should be inferred.
Nonsense, said Cicero. “ Quod si exceptio facit ne liceat, ubi non sit exceptum …” Oops, I keep forgetting how rusty folks are on subjunctives. Cicero said, if you prohibit something in certain cases, you imply that the rest of the time it’s permitted. To put it another way, the explicit statement of an exception proves that a rule to the contrary prevails otherwise.
It was a throwaway line in a two-minute video made in 2011. I’m pretty sure that no effort will be made on his part to go back on YouTube and overdub that half a sentence.
BTW, have you actually checked to see if he has said something about that error in the ten-and-a-half years since?
Okay, I might was well give you the phrase that started the discussion with my friend. I do not know what the original Latin was, but it was 19th century scholarly Latin, translated by a professor at Fordham (presumably a Jesuit). The translated phrase from Gauss’s Disquisiones Arithmeticae went something like: “Lagrange proved this by induction but I give the first demonstration by infinite descent.” Now to a modern mathematician, prove by induction and demonstrate by infinite descent have exactly the same meaning. What Lagrange did was try a whole lot of cases and so he apparently tried it over and over and inferred it by induction using that word in its non-mathematical sense. When a mathematician talks about induction, he means mathematical induction, which is identical to infinite descent.
IIRC, the proposition in question was quadratic reciprocity.
Speaking as a physicist, when I want to know if two algebraic expressions are equivalent, I pick a random value for x, and evaluate both expressions with that value. And when I say “random”, I mean “pi, unless the expression contains trig functions, and then I pick 1 instead”.
If I’m feeling really ambitious, I plug in e in both expressions, too.