Maths doesn't have hypotheses, it has conjectures and truth. This is because Maths is an axiomatic system
The axioms are the hypothesis. Change them and you get a different mathematics. In that sense, in mathematics everything is hypothetical.
Actually mathematics does have hypotheses. However there are very few, they are called axioms and hardly raise any doubt about their validity, with the possible exception of the axiom of choice.
What operators do we define as primitive and which ones do we define from those? For example we can write axioms for AND/OR/NOT in proposition logic, or we can have only axioms for the NAND operator and define the others from that.
I don't understand your argument, geometry is one of the most difficult mathematical fields, much more difficult than calculus.
Opinions and experiences on what is more difficult will be different for different people.
According to Henri Poincare (in La valeur de la science) there are two kind of mathematicians: analysts and geometers.
It is impossible to study the works of the great mathematicians, or even those of the lesser, without noticing and distinguishing two opposite tendencies, or rather two entirely different kinds of minds. The one sort are above all preoccupied with logic; to read their works, one is tempted to believe they have advanced only step by step, after the manner of a Vauban who pushes on his trenches against the place besieged, leaving nothing to chance. The other sort are guided by intuition and at the first stroke make quick but sometimes precarious conquests, like hold cavalrymen of the advance guard.
The method is not imposed by the matter treated. Though one often says of the first that they are analysts and calls the others geometers, that does
not prevent the one sort from remaining analysts even when they work at geometry, while the others are still geometers even when they occupy themselves with
pure analysis. It is the very nature of their mind which makes them logicians or intuitionalists, and they can not lay it aside when they approach a new subject. Nor
is it education which has developed in them one of the two tendencies and stifled the other. The mathematician is born, not made, and it seems he is born a geometer or an analyst.
I should like to cite examples and there are surely plenty; but to accentuate the contrast I shall begin with an extreme example, taking the liberty of seeking it in two living mathematicians. M. Meray wants to prove that a binomial equation always has a root, or, in ordinary words, that an angle may always be subdivided. If there is any truth that we think we know by direct intuition, it is this. Who could doubt that an angle may always be divided into any number of equal parts? M. Meray does not look at it that way; in his eyes this proposition is not at all evident and to prove it he needs several pages.
On the other hand, look at Professor Klein: he is studying one of the most abstract questions of the theory of functions to determine whether on a given Riemann surface there always exists a function admitting of given singularities. What does the celebrated German geometer do? He replaces his Riemann surface by a metallic surface whose electric conductivity varies according to certain laws. He connects two of its points with the two poles of a battery. The current, says he, must pass, and the distribution of this current on the surface will define a function whose singularities will be precisely those called for by the enunciation.
Doubtless Professor Klein well knows he has given here only a sketch: nevertheless he has not hesitated to publish it; and he would probably believe he finds in it, if not a rigorous demonstration, at least a kind of moral certainty. A logician would have rejected with horror such a conception, or rather he would not have
had to reject it, because in his mind it would never have originated. (PDF English translation)
AFAICT: The Wikipedia article you link to doesn't mention x86 processors at all...
I used to run a software PC Emulator on Archimedes(1) in 1987, ARM (around 4 to 8 Mips (2)) was at that time emulating 8086 at the speed of an IBM PC/XT or AT (both below 1 Mips (3)).
While calculating a screen-sized Mandelbrot fractal at the time took minutes (up to half an hour) on IBM PC, the Archimedes did it in seconds.
(1) The 80186 co-processor card mentioned at the end of the linked article was unfortunately never released, the emulation was 100% in software.
(2) Acorn Archimedes speed
(3) IBM PC/XT and AT speed
80586 / i586 was named "Pentium" because Intel could not trademark a number but still wanted to distinguish itself from AMD Am86 and Cyrix Cx486.
"80x86" has since then become a de facto generic name for all descendants of the 8086, including the x86-64 / AMD64 / EM64T / Intel64 / x64 architecture.
Setting aside that it is unclear that 100% of cartographers believed anything like that, considering that it was known since Ancient Greece that the Earth was round, a cartographer, particularly in that period, did not work under the scientific method.
I think you may be taking the comparison a bit too far: Combining actual English phrases together has significantly fewer combinations than all combinations of letters in the alphabet.
You miss the point: his algorithm is the most simple way to create all actual English (and many other languages free on top) phrases, including all the patents generated by the other generator - and also a lot of nonsense indeed. For practical use, as for generating usable prior art, both the algorithms are on par: none at all. It's an art project, not a serious attempt at solving anything.
the moon is not slowly falling into the earth. Once something has a stable orbit it tends to stay in that orbit.
That reminds me of how my first year physics professor (Roger Van Geen) explained the orbit of the moon using vectors: "it is constantly falling beside the earth."
Hi, philosopher here, teaching logic now, have been teaching maths and stats on college level before.
It is quite possible to teach stats on high school level. Stats is at basic level quite easy and useful knowledge to all: how to deal with a big heap of numbers. Summarize them to a couple of numbers which indicate how spread out they are around different kinds of central numbers. Use the numbers to make some predictions. Some combinatorics and probabilities. Have an idea what the numbers reported in the press actually mean and how reliable they are. This is to most more useful in life than calculating integrals.
So replacing calculus by stats in high school might make sense. Algebra of course is needed to understand stats (and indeed a good training for logic).
Dropping algebra for stats would make as much sense as dropping high school physics for string theory.
My kids encountered Suetonius's "The die is cast" ("alea iacta est") while still at primary school (although they didn't know at that stage it was Suetonius, only that it was said to Julius Caesar, who they thought was a character in the Asterix comic books).
Caesar died in 44BC, Suetonius was born in 69 or 70AD. So Suetonius never said anything to Caesar,
Plutarch wrote that Caesar, paraphrasing or quoting Menander, said it in Greek.
Suetonius reports Caesar saying "iacta alea est".
Obviously the universe shouldn't exist in the first place. If it does, what does it prove? That there is a god. But wait, a god shouldn't exist either.
After the above (the proves of which are generally accepted), Goedel went on to prove the existence of god (a prove which is not considered as being valid by most - it is basically a formal variant of St. Anselm's ontological argument).
Nowadays it seems pretty much all non-trivial chips include an ARM core with enough storage and ram to run circles around any desktop computer found in the 80s or even early 90s.
In fact, ARM at 4MHz (reading ROM) or 8MHz (after copying ROM into RAM) was running circles around 80s desktops in 1987 - I used to emulate an IBM PC on an Acorn Archimedes at that time.
Billions of dollars, and they can't think-up a name that sounds a bit less stupid?
'S' (for "Statistics" - originally with single quotes, those are usually being dropped now) is a programming language created in 1975-1976 at Bell Labs (which had a tradition of single letter named programming languages, such as C) on General Electrics GCOS mainframes and since 1979 on UNIX.
R is an implementation of 'S' (with lexical scoping semantics inspired by Scheme) since 1993.
Slashdot Mirror
If the file ends with "QED", then b[y] definition it is a proof.
Except of course if it is a text about Quantum Electro-Dynamics.
Maths doesn't have hypotheses, it has conjectures and truth. This is because Maths is an axiomatic system
The axioms are the hypothesis. Change them and you get a different mathematics. In that sense, in mathematics everything is hypothetical.
Actually mathematics does have hypotheses. However there are very few, they are called axioms and hardly raise any doubt about their validity, with the possible exception of the axiom of choice.
And Euclid's fifth postulate: change this hypothesis and you get different geometry.
There are other issues with axioms:
Do we restrict ourselves to first order predicate logic (like Tarski's Axioms for Euclidean geometry) or do we allow higher order logic (like Hilbert's axioms for Euclidean geometry)? Such choices have consequences for completeness and decidability.
What operators do we define as primitive and which ones do we define from those? For example we can write axioms for AND/OR/NOT in proposition logic, or we can have only axioms for the NAND operator and define the others from that.
Don't forget Digital Research's GEM, Berkeley Softworks' GEOS and VisiCorp's Visi On.
I don't understand your argument, geometry is one of the most difficult mathematical fields, much more difficult than calculus.
Opinions and experiences on what is more difficult will be different for different people.
According to Henri Poincare (in La valeur de la science) there are two kind of mathematicians: analysts and geometers.
It is impossible to study the works of the great mathematicians, or even those of the lesser, without noticing and distinguishing two opposite tendencies, or rather two entirely different kinds of minds. The one sort are above all preoccupied with logic; to read their works, one is tempted to believe they have advanced only step by step, after the manner of a Vauban who pushes on his trenches against the place besieged, leaving nothing to chance. The other sort are guided by intuition and at the first stroke make quick but sometimes precarious conquests, like hold cavalrymen of the advance guard.
The method is not imposed by the matter treated. Though one often says of the first that they are analysts and calls the others geometers, that does not prevent the one sort from remaining analysts even when they work at geometry, while the others are still geometers even when they occupy themselves with pure analysis. It is the very nature of their mind which makes them logicians or intuitionalists, and they can not lay it aside when they approach a new subject. Nor is it education which has developed in them one of the two tendencies and stifled the other. The mathematician is born, not made, and it seems he is born a geometer or an analyst.
I should like to cite examples and there are surely plenty; but to accentuate the contrast I shall begin with an extreme example, taking the liberty of seeking it in two living mathematicians. M. Meray wants to prove that a binomial equation always has a root, or, in ordinary words, that an angle may always be subdivided. If there is any truth that we think we know by direct intuition, it is this. Who could doubt that an angle may always be divided into any number of equal parts? M. Meray does not look at it that way; in his eyes this proposition is not at all evident and to prove it he needs several pages.
On the other hand, look at Professor Klein: he is studying one of the most abstract questions of the theory of functions to determine whether on a given Riemann surface there always exists a function admitting of given singularities. What does the celebrated German geometer do? He replaces his Riemann surface by a metallic surface whose electric conductivity varies according to certain laws. He connects two of its points with the two poles of a battery. The current, says he, must pass, and the distribution of this current on the surface will define a function whose singularities will be precisely those called for by the enunciation.
Doubtless Professor Klein well knows he has given here only a sketch: nevertheless he has not hesitated to publish it; and he would probably believe he finds in it, if not a rigorous demonstration, at least a kind of moral certainty. A logician would have rejected with horror such a conception, or rather he would not have had to reject it, because in his mind it would never have originated. (PDF English translation)
the site administration has lost all of the people who had any idea what DEC was.
The good thing about all this is that nobody gets confused between DEC and DR any more.
No, it would be silly to make an inconsistent language, rather it would be incomplete.
Not necessarily: paraconsistent logic
No it isn't. It is lightening your hard drive. Because it doesn't hold as much data anymore. It is now lighter.
If you had ever used punch cards, you would know that data has a negative mass.
ARM will never be able to compete with x86 in terms of computing power and x86 can't compete with ARM in terms of efficiency and low power.
Be careful with words like "never", I remember very well when ARM was running circles around 80x86 in terms of computing power: back in 1987, ARM's selling point was speed rather than low power.
AFAICT: The Wikipedia article you link to doesn't mention x86 processors at all...
I used to run a software PC Emulator on Archimedes(1) in 1987, ARM (around 4 to 8 Mips (2)) was at that time emulating 8086 at the speed of an IBM PC/XT or AT (both below 1 Mips (3)).
While calculating a screen-sized Mandelbrot fractal at the time took minutes (up to half an hour) on IBM PC, the Archimedes did it in seconds.
(1) The 80186 co-processor card mentioned at the end of the linked article was unfortunately never released, the emulation was 100% in software.
(2) Acorn Archimedes speed
(3) IBM PC/XT and AT speed
ARM will never be able to compete with x86 in terms of computing power and x86 can't compete with ARM in terms of efficiency and low power.
Be careful with words like "never", I remember very well when ARM was running circles around 80x86 in terms of computing power: back in 1987, ARM's selling point was speed rather than low power.
Intel hasn't made an 80x86 chip in a couple decades.
80586 / i586 was named "Pentium" because Intel could not trademark a number but still wanted to distinguish itself from AMD Am86 and Cyrix Cx486.
"80x86" has since then become a de facto generic name for all descendants of the 8086, including the x86-64 / AMD64 / EM64T / Intel64 / x64 architecture.
quantumness is the fact that textures are calculated from hidden variables and only instantiated when you actually look.
Nope
The new stuff has absolutely no "character" or heart in it.
The ASCII art version has plenty of characters in it.
Setting aside that it is unclear that 100% of cartographers believed anything like that, considering that it was known since Ancient Greece that the Earth was round, a cartographer, particularly in that period, did not work under the scientific method.
Don't underestimate the ancient Greek cartographers, the school of Miletus can be called the birth place of both science and philosophy, and cartography was one of the first fields where early serious attempts at science were being made.
I think you may be taking the comparison a bit too far: Combining actual English phrases together has significantly fewer combinations than all combinations of letters in the alphabet.
You miss the point: his algorithm is the most simple way to create all actual English (and many other languages free on top) phrases, including all the patents generated by the other generator - and also a lot of nonsense indeed. For practical use, as for generating usable prior art, both the algorithms are on par: none at all. It's an art project, not a serious attempt at solving anything.
In France, some groups have been generating text, graphics, architecture, history, theater, comics, etc... in a systematical way since 1960.
the moon is not slowly falling into the earth. Once something has a stable orbit it tends to stay in that orbit.
That reminds me of how my first year physics professor (Roger Van Geen) explained the orbit of the moon using vectors: "it is constantly falling beside the earth."
I'm interested in a real life example
LabNation SmartScope software is (partly) written using Xamarin, it runs on Linux, MacOS, Windows, Android and iOS.
Link to GitHub.
Surely robots only need 3 Laws?
R. Daneel Olivaw disagrees.
Hi, philosopher here, teaching logic now, have been teaching maths and stats on college level before.
It is quite possible to teach stats on high school level. Stats is at basic level quite easy and useful knowledge to all: how to deal with a big heap of numbers. Summarize them to a couple of numbers which indicate how spread out they are around different kinds of central numbers. Use the numbers to make some predictions. Some combinatorics and probabilities. Have an idea what the numbers reported in the press actually mean and how reliable they are. This is to most more useful in life than calculating integrals.
So replacing calculus by stats in high school might make sense. Algebra of course is needed to understand stats (and indeed a good training for logic).
Dropping algebra for stats would make as much sense as dropping high school physics for string theory.
My kids encountered Suetonius's "The die is cast" ("alea iacta est") while still at primary school (although they didn't know at that stage it was Suetonius, only that it was said to Julius Caesar, who they thought was a character in the Asterix comic books).
Caesar died in 44BC, Suetonius was born in 69 or 70AD. So Suetonius never said anything to Caesar,
Plutarch wrote that Caesar, paraphrasing or quoting Menander, said it in Greek. Suetonius reports Caesar saying "iacta alea est".
[...] (the proofs of which are generally accepted), [...] (a proof which is not considered as being valid by most [...] ).
Fixed that For Me.
Wasn't there some guy named Goedel who theorized something about systems being unable to prove their own consistency?
In his 1929 PhD dissertation, Goedel showed that first order logic can prove both its own consistency and completeness - so systems which can prove their own consistency do exist.
The famous 1931 "Incompleteness Theorems of Goedel" proved that second (and higher) order logic cannot be consistent and complete at the same time. Most mathematics, especially at the time, needed second order logic. In the mean time, a lot of maths have been recreated in first order logic, by Tarski and others.
Obviously the universe shouldn't exist in the first place. If it does, what does it prove? That there is a god. But wait, a god shouldn't exist either.
After the above (the proves of which are generally accepted), Goedel went on to prove the existence of god (a prove which is not considered as being valid by most - it is basically a formal variant of St. Anselm's ontological argument).
Nowadays it seems pretty much all non-trivial chips include an ARM core with enough storage and ram to run circles around any desktop computer found in the 80s or even early 90s.
In fact, ARM at 4MHz (reading ROM) or 8MHz (after copying ROM into RAM) was running circles around 80s desktops in 1987 - I used to emulate an IBM PC on an Acorn Archimedes at that time.
Billions of dollars, and they can't think-up a name that sounds a bit less stupid?
'S' (for "Statistics" - originally with single quotes, those are usually being dropped now) is a programming language created in 1975-1976 at Bell Labs (which had a tradition of single letter named programming languages, such as C) on General Electrics GCOS mainframes and since 1979 on UNIX.
R is an implementation of 'S' (with lexical scoping semantics inspired by Scheme) since 1993.
Respectfully, it would be wise to read a Statistics 101 handbook.
You invoked the gambler's fallacy, so it was you who assumed a fair coin in the first place.
The calculations I did, and the point of the post by Pulzar you were originally replying to were showing that it is not a fair coin.
You obviously do not understand probabilities.