mitchr...@gmail.com wrote :
> On Friday, May 14, 2021 at 11:41:32 AM UTC-7, Michael Moroney wrote:
>> On 5/14/2021 2:28 PM, mitchr...@gmail.com wrote:
>>> The imaginary isn't real because of its negative.
>>> You cannot subject a subtraction to a root...
>> But you certainly subject a negative number to a root function.
>
> No. That does not have an answer. That is why they chose
> the name.

Inverting the odd exponents worked fine.

On Friday, May 14, 2021 at 5:13:22 PM UTC-7, FromTheRafters wrote:
> mitchr...@gmail.com wrote :
> > On Friday, May 14, 2021 at 11:41:32 AM UTC-7, Michael Moroney wrote:
> >> On 5/14/2021 2:28 PM, mitchr...@gmail.com wrote:
> >>> The imaginary isn't real because of its negative.
> >>> You cannot subject a subtraction to a root...
> >> But you certainly subject a negative number to a root function.
> >
> > No. That does not have an answer. That is why they chose
> > the name.
> Inverting the odd exponents worked fine.

So what do they do you moron?
Show that you mean something instead...

Mitchell Raemsch

On 5/14/2021 6:57 PM, mitchr...@gmail.com wrote:
> On Friday, May 14, 2021 at 11:41:32 AM UTC-7, Michael Moroney wrote:
>> On 5/14/2021 2:28 PM, mitchr...@gmail.com wrote:
>>> The imaginary isn't real because of its negative.
>>> You cannot subject a subtraction to a root...
>> But you certainly subject a negative number to a root function.
>
> No. That does not have an answer.

Of course it is an answer, Roy Masters! As long as you are open to the
fact the answer is not a real number.

> That is why they chose
> the name. Were mathematicians wrong you moron?
>
Only you are wrong, Smitch. Every single time.

On Friday, 14 May 2021 at 19:46:15 UTC-4, Eram semper recta wrote:
> On Friday, 14 May 2021 at 19:42:04 UTC-4, Eram semper recta wrote:
> > On Friday, 14 May 2021 at 18:02:49 UTC-4, Eram semper recta wrote:
> > > On Friday, 14 May 2021 at 14:08:42 UTC-4, ju...@diegidio.name wrote:
> > > > On Friday, 14 May 2021 at 14:56:26 UTC+2, Eram semper recta wrote:
> > > > > On Friday, 14 May 2021 at 08:52:44 UTC-4, Eram semper recta wrote:
> > > > > > On Friday, 14 May 2021 at 08:00:30 UTC-4, ju...@diegidio.name wrote:
> > > >
> > > > > > > That the diagonal of a square is incommensurable to its side was a scandal at the time, some 2500 years ago...
> > > >
> > > > > In fact that statement is also only half true because the diagonal of a square is incommensurable to ANY other magnitude, not just the side of a square!
> > > > That's plain wrong, and precisely the measure of what a crank you are: to begin with, and for the millionth time, open up any decent dictionary and realise that _incommensurable_ is of two things relative to one another... crank.
> > > Hey moron. I have been over this with you before. You didn't get it then, so what makes you think you'll get it now?
> > >
> > > > EOD.
> > >
> > > There never was a discussion, you pathetic buffoon! I am ridiculing your stupidity and ignorance. It is you who has no clue what *incommensurable* means!
> > >
> > > We say that p and q are commensurable with each other if and only if they both have a common divisor. Did you get this, you dumb ape?
> > By common divisor, I mean any part of a given measure or the measure itself that divides both p and q.
> >
> > The word commensurable comes from a Greek word meaning "same measure".
> >
> > You might understand this some day. But I hold out little hope you will any time soon.
> commensurable: p and q divisible without remainder by a common unit
>
> But we are way off topic here.
>
> Zelos Malum, the Swede crank who claims to have a Master's in math cannot explain what is meant by multiplication and cannot produce a definition that applies to all numbers.
>
> All operations of arithmetic can be stated strictly in terms of subtraction. Addition is derived immediately from subtraction.
>
> Division:
>
> A quotient ps/qr is the rational number that is measured in terms of two numbers p/q and r/s.
>
> ps/qr = [ p/q +p/q + ...n times ] / [ r/s + r/s+... n times ] where n=q or n=s.
>
>
> Example: 2/3 -:- 5/6 = 4/5
>
> p/q = 2/3 and r/s = 5/6
>
> 2/3 -:- 5/6 = [ 2/3 + 2/3 + 2/3 ] / [ 5/6 + 5/6 + 5/6] = [2] / [ 5/2] = [2 + 2] / [5/2 + 5/2] = 4 / 5
>
> After the second equals sign, cancellation is again possible by Book 5, Proposition 12.
>
> So the ENTIRE operation is done through geometry using similar triangles. No numbers are even required in the geometric definition, only magnitudes.
>
>
> Multiplication:
>
> The product (or multiplication) of two numbers p/q and r/s, that is, pr/qs is the quotient of either number
> with the reciprocal of the other.
>
> pr/qs = [ p/q +p/q + ...n times ] / [ s/r + s/r+... n times ] where n=q or n=r.
>
> Example: 2/3 x 5/6 = 5/9
>
> p/q = 2/3 and s/r = 6/5
>
> 2/3 x 5/6 = [2/3+2/3+2/3] / [6/5+6/5+6/5] = [2] / [18/5] = [ 2+2+2+2+2 ] / [18/5+18/5+18/5+18/5+18/5] = 10 / 18 = 5/9
>
> AND
>
> pr/qs = [ r/s + r/s+... n times ] / [ s/p + s/p+... n times ] where n=p or n=s.
>
> This part is left as an exercise.
>
> NOTE: The above definitions work on ANY magnitude. There is no need to have a separate interpretation for integers and fractions.
>
> All these details are explained in my world famous article:
>
> https://drive.google.com/file/d/0B-mOEooW03iLYTg1TGY4RTIwakU
>
> Compare this clarity with the bullshit of Zelos Malum:
>
> ------------------------------------------------------------------------
> i. a/b=[(a,b)]
> ii. [(a,b)][(c,d)]=[(ac,bd)]
> iii. 2.1=21/10=[(21,10)]
> iv. 2.2=22/10=[(22,10)]
> v. 2.1*2.2=[(21,10)][(22,10)]=[(462,100)]=4.62
> -------------------------------------------------------------------------

It seems that Zelos Malum has finally realised he is a moron. Chuckle.

Not only does he fail from the first step in which he believes division is possible since to him a/b means "a divided by b" and multiplication is derived from division, but he assumes multiplication in step 2.

Well, perhaps some day there'll be a mainstream moron who might admit the mainstream was and is WRONG about so many things in mathematics.

lördag 15 maj 2021 kl. 01:46:15 UTC+2 skrev Eram semper recta:
> On Friday, 14 May 2021 at 19:42:04 UTC-4, Eram semper recta wrote:
> > On Friday, 14 May 2021 at 18:02:49 UTC-4, Eram semper recta wrote:
> > > On Friday, 14 May 2021 at 14:08:42 UTC-4, ju...@diegidio.name wrote:
> > > > On Friday, 14 May 2021 at 14:56:26 UTC+2, Eram semper recta wrote:
> > > > > On Friday, 14 May 2021 at 08:52:44 UTC-4, Eram semper recta wrote:
> > > > > > On Friday, 14 May 2021 at 08:00:30 UTC-4, ju...@diegidio.name wrote:
> > > >
> > > > > > > That the diagonal of a square is incommensurable to its side was a scandal at the time, some 2500 years ago...
> > > >
> > > > > In fact that statement is also only half true because the diagonal of a square is incommensurable to ANY other magnitude, not just the side of a square!
> > > > That's plain wrong, and precisely the measure of what a crank you are: to begin with, and for the millionth time, open up any decent dictionary and realise that _incommensurable_ is of two things relative to one another... crank.
> > > Hey moron. I have been over this with you before. You didn't get it then, so what makes you think you'll get it now?
> > >
> > > > EOD.
> > >
> > > There never was a discussion, you pathetic buffoon! I am ridiculing your stupidity and ignorance. It is you who has no clue what *incommensurable* means!
> > >
> > > We say that p and q are commensurable with each other if and only if they both have a common divisor. Did you get this, you dumb ape?
> > By common divisor, I mean any part of a given measure or the measure itself that divides both p and q.
> >
> > The word commensurable comes from a Greek word meaning "same measure".
> >
> > You might understand this some day. But I hold out little hope you will any time soon.
> commensurable: p and q divisible without remainder by a common unit
>
> But we are way off topic here.
>
> Zelos Malum, the Swede crank who claims to have a Master's in math cannot explain what is meant by multiplication and cannot produce a definition that applies to all numbers.
> All operations of arithmetic can be stated strictly in terms of subtraction. Addition is derived immediately from subtraction.
>
> Division:
>
> A quotient ps/qr is the rational number that is measured in terms of two numbers p/q and r/s.
>
> ps/qr = [ p/q +p/q + ...n times ] / [ r/s + r/s+... n times ] where n=q or n=s.
>
>
> Example: 2/3 -:- 5/6 = 4/5
>
> p/q = 2/3 and r/s = 5/6
>
> 2/3 -:- 5/6 = [ 2/3 + 2/3 + 2/3 ] / [ 5/6 + 5/6 + 5/6] = [2] / [ 5/2] = [2 + 2] / [5/2 + 5/2] = 4 / 5
>
> After the second equals sign, cancellation is again possible by Book 5, Proposition 12.
>
> So the ENTIRE operation is done through geometry using similar triangles. No numbers are even required in the geometric definition, only magnitudes.
>
>
> Multiplication:
>
> The product (or multiplication) of two numbers p/q and r/s, that is, pr/qs is the quotient of either number
> with the reciprocal of the other.
>
> pr/qs = [ p/q +p/q + ...n times ] / [ s/r + s/r+... n times ] where n=q or n=r.
>
> Example: 2/3 x 5/6 = 5/9
>
> p/q = 2/3 and s/r = 6/5
>
> 2/3 x 5/6 = [2/3+2/3+2/3] / [6/5+6/5+6/5] = [2] / [18/5] = [ 2+2+2+2+2 ] / [18/5+18/5+18/5+18/5+18/5] = 10 / 18 = 5/9
>
> AND
>
> pr/qs = [ r/s + r/s+... n times ] / [ s/p + s/p+... n times ] where n=p or n=s.
>
> This part is left as an exercise.
>
> NOTE: The above definitions work on ANY magnitude. There is no need to have a separate interpretation for integers and fractions.
>
> All these details are explained in my world famous article:
>
> https://drive.google.com/file/d/0B-mOEooW03iLYTg1TGY4RTIwakU
>
> Compare this clarity with the bullshit of Zelos Malum:
>
> ------------------------------------------------------------------------
> i. a/b=[(a,b)]
> ii. [(a,b)][(c,d)]=[(ac,bd)]
> iii. 2.1=21/10=[(21,10)]
> iv. 2.2=22/10=[(22,10)]
> v. 2.1*2.2=[(21,10)][(22,10)]=[(462,100)]=4.62
> -------------------------------------------------------------------------

Each number construction has its own definition based on what it uses prior.. There is no universal in anything there.

natural numbers have one, integers have one, rational numbers have eone, real have one, complex has one, and so on and so on.

onsdag 12 maj 2021 kl. 15:12:59 UTC+2 skrev Eram semper recta:
> On Wednesday, 12 May 2021 at 01:00:01 UTC-4, zelos...@gmail.com wrote:
> > >Poor MALUM. Just repeating lies won't make them true. Tsk, tsk.
> > You gave NOTHING for the Eudoxus one, you just harped on "measure" but you never ONCE adressed the definitions! You are the one lying.
> Eudoxus of Cnidus used only **magnitudes** and that bullshit construction in the pdf you provided tries to build on his ideas albeit unsuccessfully.
> > >I might be inclined to let you go with a/b=[(a,b)] but that's about all.
> > that is the notational definition
> > >In step 2, ie, [(a,b)][(c,d)]=[(ac,bd)] , you've already assumed that ac is the product of a x c.
>
> > no, ac is shorthand writing for a x c, there is no assumption. Everyone knows that ac=a*c=a x c
> <big boring rant by Malum>
>
> If we appeared in front of a knowledgeable judge as attorneys, I would have responded:
>
> "Your honour, I rest my case."
>
> Even the dumbest on this newsgroup know that you are an idiot. You were asked to demonstrate what that operator x (multiplication) means. Not only did you fail dismally in assuming that it means ac, but you failed to understand the question!
>
> So here it is again for your 13th try? Chuckle.
> In geometry, multiplication is defined as the *reciprocal measure* of division.
> Formal definition in my world famous article called How we got numbers:
> A quotient (or division) is that positive rational number, that is measured in terms of two numbers. In its most primitive form, a quotient is simply the ratio of two magnitudes which both share a common measure (divisor).
>
> So we actually determine 21/10 x 11/5 either as:
>
> i. 21/10 -:- 5/11
> ii. 11/5 -:- 10/21
>
> I'll show you (i) and leave (ii) as an exercise.
>
> Using similar triangles from which ALL the arithmetic operations are defined we proceed as follows:
>
> (21+21+21+21+21+21+21+21+21+21+21) / (10+10+10+10+10+10+10+10+10+10+10)
> /
> (5 + 5 + 5 +5 + 5 + 5 + 5 + 5 + 5 +5 ) / (11+11+11+11+11+11+11+11+11+11)
> = (231/110) / (50/110)
> = 231/50
>
> So, we find the product using the proportional sides of similar triangles. Now since 231 > 50, we can apply the obelus division algorithm, ie
> 231 -:- 50 = 4.62
>
> 231 - 50 = 181
> 181 - 50 = 131
> 131 - 50 = 81
> 81 - 50 = 31
>
> We have reached the stopping condition, ie 31 (remainder) < 50 (divisor). Since we subtracted four times, the partial quotient is 4 but if we add the remainder, then we get the full quotient 4 + 31/50 = 4.62
>
> Again, my article explains all these details:
>
> https://drive.google.com/file/d/0B-mOEooW03iLYTg1TGY4RTIwakU
> You see Malum, what you scribbled is like a tiny Pigeon's wet fart, not even smelly at that! LMAO.
> On the other hand what I have shown you is that multiplication was defined LONG before number was realised and that its properties were transferred to algebra (a weaker form of geometry) via the ABSTRACT UNIT.
Notice you still haven't shown any internal error in any of them, especially with eudoxus :)

On Monday, 17 May 2021 at 01:11:26 UTC-4, zelos...@gmail.com wrote:
> >Eudoxus of Cnidus used only **magnitudes** and that bullshit construction in the pdf you provided tries to build on his ideas albeit unsuccessfully.
>
> Who cares? We are not talking about his work.
>
> >Even the dumbest on this newsgroup know that you are an idiot. You were asked to demonstrate what that operator x (multiplication) means. Not only did you fail dismally in assuming that it means ac, but you failed to understand the question!
>
> There it is the multiplication of integers, duh.

Newsflash: Integers are RATIONAL NUMBERS.

Eyes rolling and FACEPALM.

On Monday, 17 May 2021 at 01:12:33 UTC-4, zelos...@gmail.com wrote:
> >So when Zelos is asked to actually show why he thinks a certain statement is true, he imagines that he can just define an operator without explaining how it works. This is the typical retort that stupid math educators produce when they are questioned by students and have no answers.
>
> The question was define rationals from integers, I am free to assume the integer stuff is done then :)

Again, integers are rational numbers. The FACT that you can multiply numerators by numerators and denominators by denominators is a consequence of what is transferred to algebra FROM GEOMETRY, you utter moron.

You show with every post, your pitiful ineptitude to understand not only mathematics, but also the rudiments of English grammar.

On Monday, 17 May 2021 at 01:14:47 UTC-4, zelos...@gmail.com wrote:
> lördag 15 maj 2021 kl. 01:46:15 UTC+2 skrev Eram semper recta:
> > On Friday, 14 May 2021 at 19:42:04 UTC-4, Eram semper recta wrote:
> > > On Friday, 14 May 2021 at 18:02:49 UTC-4, Eram semper recta wrote:
> > > > On Friday, 14 May 2021 at 14:08:42 UTC-4, ju...@diegidio.name wrote:
> > > > > On Friday, 14 May 2021 at 14:56:26 UTC+2, Eram semper recta wrote:
> > > > > > On Friday, 14 May 2021 at 08:52:44 UTC-4, Eram semper recta wrote:
> > > > > > > On Friday, 14 May 2021 at 08:00:30 UTC-4, ju...@diegidio.name wrote:
> > > > >
> > > > > > > > That the diagonal of a square is incommensurable to its side was a scandal at the time, some 2500 years ago...
> > > > >
> > > > > > In fact that statement is also only half true because the diagonal of a square is incommensurable to ANY other magnitude, not just the side of a square!
> > > > > That's plain wrong, and precisely the measure of what a crank you are: to begin with, and for the millionth time, open up any decent dictionary and realise that _incommensurable_ is of two things relative to one another... crank.
> > > > Hey moron. I have been over this with you before. You didn't get it then, so what makes you think you'll get it now?
> > > >
> > > > > EOD.
> > > >
> > > > There never was a discussion, you pathetic buffoon! I am ridiculing your stupidity and ignorance. It is you who has no clue what *incommensurable* means!
> > > >
> > > > We say that p and q are commensurable with each other if and only if they both have a common divisor. Did you get this, you dumb ape?
> > > By common divisor, I mean any part of a given measure or the measure itself that divides both p and q.
> > >
> > > The word commensurable comes from a Greek word meaning "same measure"..
> > >
> > > You might understand this some day. But I hold out little hope you will any time soon.
> > commensurable: p and q divisible without remainder by a common unit
> >
> > But we are way off topic here.
> >
> > Zelos Malum, the Swede crank who claims to have a Master's in math cannot explain what is meant by multiplication and cannot produce a definition that applies to all numbers.
> > All operations of arithmetic can be stated strictly in terms of subtraction. Addition is derived immediately from subtraction.
> >
> > Division:
> >
> > A quotient ps/qr is the rational number that is measured in terms of two numbers p/q and r/s.
> >
> > ps/qr = [ p/q +p/q + ...n times ] / [ r/s + r/s+... n times ] where n=q or n=s.
> >
> >
> > Example: 2/3 -:- 5/6 = 4/5
> >
> > p/q = 2/3 and r/s = 5/6
> >
> > 2/3 -:- 5/6 = [ 2/3 + 2/3 + 2/3 ] / [ 5/6 + 5/6 + 5/6] = [2] / [ 5/2] = [2 + 2] / [5/2 + 5/2] = 4 / 5
> >
> > After the second equals sign, cancellation is again possible by Book 5, Proposition 12.
> >
> > So the ENTIRE operation is done through geometry using similar triangles. No numbers are even required in the geometric definition, only magnitudes.
> >
> >
> > Multiplication:
> >
> > The product (or multiplication) of two numbers p/q and r/s, that is, pr/qs is the quotient of either number
> > with the reciprocal of the other.
> >
> > pr/qs = [ p/q +p/q + ...n times ] / [ s/r + s/r+... n times ] where n=q or n=r.
> >
> > Example: 2/3 x 5/6 = 5/9
> >
> > p/q = 2/3 and s/r = 6/5
> >
> > 2/3 x 5/6 = [2/3+2/3+2/3] / [6/5+6/5+6/5] = [2] / [18/5] = [ 2+2+2+2+2 ] / [18/5+18/5+18/5+18/5+18/5] = 10 / 18 = 5/9
> >
> > AND
> >
> > pr/qs = [ r/s + r/s+... n times ] / [ s/p + s/p+... n times ] where n=p or n=s.
> >
> > This part is left as an exercise.
> >
> > NOTE: The above definitions work on ANY magnitude. There is no need to have a separate interpretation for integers and fractions.
> >
> > All these details are explained in my world famous article:
> >
> > https://drive.google.com/file/d/0B-mOEooW03iLYTg1TGY4RTIwakU
> >
> > Compare this clarity with the bullshit of Zelos Malum:
> >
> > ------------------------------------------------------------------------
> > i. a/b=[(a,b)]
> > ii. [(a,b)][(c,d)]=[(ac,bd)]
> > iii. 2.1=21/10=[(21,10)]
> > iv. 2.2=22/10=[(22,10)]
> > v. 2.1*2.2=[(21,10)][(22,10)]=[(462,100)]=4.62
> > -------------------------------------------------------------------------
> Each number construction has its own definition based on what it uses prior.

FALSE. Every number (without exception) is a RATIONAL NUMBER and the arithmetic operations are transferred from GEOMETRY to ALGEBRA applicable ONLY to RATIONAL NUMBERS. In geometry we can multiply pi by sqrt2 exactly based on a given UNIT. In algebra, we cannot do this with the ABSTRACT UNIT. Try to understand, you pathetic crank!

> There is no universal in anything there.

There MOST definitely IS a general definition as I have PROVED and so far all you have done is whine.

We are still waiting for you to show how p/q x r/s actually works in your BOGUS set theory definitions if you can even call them that because your definitions are CIRCULAR in every respect.

>
> natural numbers have one, integers have one, rational numbers have eone, real have one, complex has one, and so on and so on.

NO, NO, NO. And again, NO.

Natural numbers and integers ARE RATIONAL NUMBERS, you fucking moron!!!! These are just a special case of the GENERAL definition which is RATIONAL NUMBER. Get it crank? LMAO.

On Monday, 17 May 2021 at 01:15:51 UTC-4, zelos...@gmail.com wrote:
> onsdag 12 maj 2021 kl. 15:12:59 UTC+2 skrev Eram semper recta:
> > On Wednesday, 12 May 2021 at 01:00:01 UTC-4, zelos...@gmail.com wrote:
> > > >Poor MALUM. Just repeating lies won't make them true. Tsk, tsk.
> > > You gave NOTHING for the Eudoxus one, you just harped on "measure" but you never ONCE adressed the definitions! You are the one lying.
> > Eudoxus of Cnidus used only **magnitudes** and that bullshit construction in the pdf you provided tries to build on his ideas albeit unsuccessfully.
> > > >I might be inclined to let you go with a/b=[(a,b)] but that's about all.
> > > that is the notational definition
> > > >In step 2, ie, [(a,b)][(c,d)]=[(ac,bd)] , you've already assumed that ac is the product of a x c.
> >
> > > no, ac is shorthand writing for a x c, there is no assumption. Everyone knows that ac=a*c=a x c
> > <big boring rant by Malum>
> >
> > If we appeared in front of a knowledgeable judge as attorneys, I would have responded:
> >
> > "Your honour, I rest my case."
> >
> > Even the dumbest on this newsgroup know that you are an idiot. You were asked to demonstrate what that operator x (multiplication) means. Not only did you fail dismally in assuming that it means ac, but you failed to understand the question!
> >
> > So here it is again for your 13th try? Chuckle.
> > In geometry, multiplication is defined as the *reciprocal measure* of division.
> > Formal definition in my world famous article called How we got numbers:
> > A quotient (or division) is that positive rational number, that is measured in terms of two numbers. In its most primitive form, a quotient is simply the ratio of two magnitudes which both share a common measure (divisor)..
> >
> > So we actually determine 21/10 x 11/5 either as:
> >
> > i. 21/10 -:- 5/11
> > ii. 11/5 -:- 10/21
> >
> > I'll show you (i) and leave (ii) as an exercise.
> >
> > Using similar triangles from which ALL the arithmetic operations are defined we proceed as follows:
> >
> > (21+21+21+21+21+21+21+21+21+21+21) / (10+10+10+10+10+10+10+10+10+10+10)
> > /
> > (5 + 5 + 5 +5 + 5 + 5 + 5 + 5 + 5 +5 ) / (11+11+11+11+11+11+11+11+11+11)
> > = (231/110) / (50/110)
> > = 231/50
> >
> > So, we find the product using the proportional sides of similar triangles. Now since 231 > 50, we can apply the obelus division algorithm, ie
> > 231 -:- 50 = 4.62
> >
> > 231 - 50 = 181
> > 181 - 50 = 131
> > 131 - 50 = 81
> > 81 - 50 = 31
> >
> > We have reached the stopping condition, ie 31 (remainder) < 50 (divisor). Since we subtracted four times, the partial quotient is 4 but if we add the remainder, then we get the full quotient 4 + 31/50 = 4.62
> >
> > Again, my article explains all these details:
> >
> > https://drive.google.com/file/d/0B-mOEooW03iLYTg1TGY4RTIwakU
> > You see Malum, what you scribbled is like a tiny Pigeon's wet fart, not even smelly at that! LMAO.
> > On the other hand what I have shown you is that multiplication was defined LONG before number was realised and that its properties were transferred to algebra (a weaker form of geometry) via the ABSTRACT UNIT.
> Notice you still haven't shown any internal error in any of them, especially with eudoxus :)

See, there you go again.... Chuckle. The bullshit Eudoxus definition as I have demonstrated to you is based on MAGNITUDES and extrapolated to your delusional conclusions about real numbers. There is nothing in Arfan's PDF that even resembles rational thought. It is BULLSHIT in the simplest terms. Only deluded, uneducated, ignorant teacher sycophants like you fall for that kind of drivel. LMAO.

On Monday, 17 May 2021 at 07:37:04 UTC-4, Eram semper recta wrote:
> On Monday, 17 May 2021 at 01:14:47 UTC-4, zelos...@gmail.com wrote:
> > lördag 15 maj 2021 kl. 01:46:15 UTC+2 skrev Eram semper recta:
> > > On Friday, 14 May 2021 at 19:42:04 UTC-4, Eram semper recta wrote:
> > > > On Friday, 14 May 2021 at 18:02:49 UTC-4, Eram semper recta wrote:
> > > > > On Friday, 14 May 2021 at 14:08:42 UTC-4, ju...@diegidio.name wrote:
> > > > > > On Friday, 14 May 2021 at 14:56:26 UTC+2, Eram semper recta wrote:
> > > > > > > On Friday, 14 May 2021 at 08:52:44 UTC-4, Eram semper recta wrote:
> > > > > > > > On Friday, 14 May 2021 at 08:00:30 UTC-4, ju...@diegidio.name wrote:
> > > > > >
> > > > > > > > > That the diagonal of a square is incommensurable to its side was a scandal at the time, some 2500 years ago...
> > > > > >
> > > > > > > In fact that statement is also only half true because the diagonal of a square is incommensurable to ANY other magnitude, not just the side of a square!
> > > > > > That's plain wrong, and precisely the measure of what a crank you are: to begin with, and for the millionth time, open up any decent dictionary and realise that _incommensurable_ is of two things relative to one another... crank.
> > > > > Hey moron. I have been over this with you before. You didn't get it then, so what makes you think you'll get it now?
> > > > >
> > > > > > EOD.
> > > > >
> > > > > There never was a discussion, you pathetic buffoon! I am ridiculing your stupidity and ignorance. It is you who has no clue what *incommensurable* means!
> > > > >
> > > > > We say that p and q are commensurable with each other if and only if they both have a common divisor. Did you get this, you dumb ape?
> > > > By common divisor, I mean any part of a given measure or the measure itself that divides both p and q.
> > > >
> > > > The word commensurable comes from a Greek word meaning "same measure".
> > > >
> > > > You might understand this some day. But I hold out little hope you will any time soon.
> > > commensurable: p and q divisible without remainder by a common unit
> > >
> > > But we are way off topic here.
> > >
> > > Zelos Malum, the Swede crank who claims to have a Master's in math cannot explain what is meant by multiplication and cannot produce a definition that applies to all numbers.
> > > All operations of arithmetic can be stated strictly in terms of subtraction. Addition is derived immediately from subtraction.
> > >
> > > Division:
> > >
> > > A quotient ps/qr is the rational number that is measured in terms of two numbers p/q and r/s.
> > >
> > > ps/qr = [ p/q +p/q + ...n times ] / [ r/s + r/s+... n times ] where n=q or n=s.
> > >
> > >
> > > Example: 2/3 -:- 5/6 = 4/5
> > >
> > > p/q = 2/3 and r/s = 5/6
> > >
> > > 2/3 -:- 5/6 = [ 2/3 + 2/3 + 2/3 ] / [ 5/6 + 5/6 + 5/6] = [2] / [ 5/2] = [2 + 2] / [5/2 + 5/2] = 4 / 5
> > >
> > > After the second equals sign, cancellation is again possible by Book 5, Proposition 12.
> > >
> > > So the ENTIRE operation is done through geometry using similar triangles. No numbers are even required in the geometric definition, only magnitudes.
> > >
> > >
> > > Multiplication:
> > >
> > > The product (or multiplication) of two numbers p/q and r/s, that is, pr/qs is the quotient of either number
> > > with the reciprocal of the other.
> > >
> > > pr/qs = [ p/q +p/q + ...n times ] / [ s/r + s/r+... n times ] where n=q or n=r.
> > >
> > > Example: 2/3 x 5/6 = 5/9
> > >
> > > p/q = 2/3 and s/r = 6/5
> > >
> > > 2/3 x 5/6 = [2/3+2/3+2/3] / [6/5+6/5+6/5] = [2] / [18/5] = [ 2+2+2+2+2 ] / [18/5+18/5+18/5+18/5+18/5] = 10 / 18 = 5/9
> > >
> > > AND
> > >
> > > pr/qs = [ r/s + r/s+... n times ] / [ s/p + s/p+... n times ] where n=p or n=s.
> > >
> > > This part is left as an exercise.
> > >
> > > NOTE: The above definitions work on ANY magnitude. There is no need to have a separate interpretation for integers and fractions.
> > >
> > > All these details are explained in my world famous article:
> > >
> > > https://drive.google.com/file/d/0B-mOEooW03iLYTg1TGY4RTIwakU
> > >
> > > Compare this clarity with the bullshit of Zelos Malum:
> > >
> > > ------------------------------------------------------------------------
> > > i. a/b=[(a,b)]
> > > ii. [(a,b)][(c,d)]=[(ac,bd)]
> > > iii. 2.1=21/10=[(21,10)]
> > > iv. 2.2=22/10=[(22,10)]
> > > v. 2.1*2.2=[(21,10)][(22,10)]=[(462,100)]=4.62
> > > -------------------------------------------------------------------------
> > Each number construction has its own definition based on what it uses prior.
> FALSE. Every number (without exception) is a RATIONAL NUMBER and the arithmetic operations are transferred from GEOMETRY to ALGEBRA applicable ONLY to RATIONAL NUMBERS. In geometry we can multiply pi by sqrt2 exactly based on a given UNIT. In algebra, we cannot do this with the ABSTRACT UNIT. Try to understand, you pathetic crank!
> > There is no universal in anything there.
> There MOST definitely IS a general definition as I have PROVED and so far all you have done is whine.
>
> We are still waiting for you to show how p/q x r/s actually works in your BOGUS set theory definitions if you can even call them that because your definitions are CIRCULAR in every respect.
> >
> > natural numbers have one, integers have one, rational numbers have eone, real have one, complex has one, and so on and so on.
> NO, NO, NO. And again, NO.
>
> Natural numbers and integers ARE RATIONAL NUMBERS, you fucking moron!!!! These are just a special case of the GENERAL definition which is RATIONAL NUMBER. Get it crank? LMAO.

Rational numbers came long, long, long before natural numbers and integers. If you had studied Euclid's Elements, then you might have known these things.

On Monday, May 17, 2021 at 7:30:01 AM UTC-4, Eram semper recta wrote:
> On Monday, 17 May 2021 at 01:11:26 UTC-4, zelos...@gmail.com wrote:
> > >Eudoxus of Cnidus used only **magnitudes** and that bullshit construction in the pdf you provided tries to build on his ideas albeit unsuccessfully.
> >
> > Who cares? We are not talking about his work.
> >
> > >Even the dumbest on this newsgroup know that you are an idiot. You were asked to demonstrate what that operator x (multiplication) means. Not only did you fail dismally in assuming that it means ac, but you failed to understand the question!
> >
> > There it is the multiplication of integers, duh.
> Newsflash: Integers are RATIONAL NUMBERS.
>
> Eyes rolling and FACEPALM.

This is a perfect instance of mathematical inversion of structure. Can it be stated that the rational numbers are constructed out of two integers? Should it then be stated that integers can be constructed out of rational numbers? Set theory allows elements to be sets, and allows sets to swallow up other sets, and there is a lack of structural integrity as a result. This is like the finger pointing game inside of a disfunctional capitalist business: the software engineer points his finger at the hardware engineer, while the hardware engineer points his finger at the software engineer and both are right. Meanwhile a bug in the firmware that some third party wrote goes undiscovered... like a magician waiving his hand around in a large sweeping motion... just as a professor at a chalkboard would do, then rapidly moving on to some other conundrum that seems more deeply troubling... the toughest bugs are built within the assumptions that we operate upon. Their extent can go rather far this way and pop out in the strangest places as side effects in a corrupted system. That we exist in such a corrupted system has to be understood as a possibility within mathematical analysis. What seems like a digression to the status quo who rides somehow and sloppily atop such trouble as if they have arrived... these are the straight A's; they can run a tight ship and they know too well that loose lips sink ships.

Timothy Golden pretended :
> On Monday, May 17, 2021 at 7:30:01 AM UTC-4, Eram semper recta wrote:
>> On Monday, 17 May 2021 at 01:11:26 UTC-4, zelos...@gmail.com wrote:
>>>> Eudoxus of Cnidus used only **magnitudes** and that bullshit construction
>>>> in the pdf you provided tries to build on his ideas albeit unsuccessfully.
>>>
>>> Who cares? We are not talking about his work.
>>>
>>>> Even the dumbest on this newsgroup know that you are an idiot. You were
>>>> asked to demonstrate what that operator x (multiplication) means. Not only
>>>> did you fail dismally in assuming that it means ac, but you failed to
>>>> understand the question!
>>>
>>> There it is the multiplication of integers, duh.
>> Newsflash: Integers are RATIONAL NUMBERS.
>>
>> Eyes rolling and FACEPALM.
>
> This is a perfect instance of mathematical inversion of structure. Can it be
> stated that the rational numbers are constructed out of two integers?

Ordered pairs of integers, yes. The integers themselves can also be
considered as ordered pairs under this construction and retain their
features such as their algebraic structure. It is an embedding of
naturals into the integers (ordered pairs of naturals) and integers
into rationals (ordered pairs of integers no zero denominators) - but
this wasn't good enough, we needed another way to think of numbers
which included irrational numbers so we leave no gaps to fill like we
did with the naturals, integers, and rationals.

So we have another way to think of number where we describe what it is
"not" -- no greater than this value, no less than this value.
Computable numbers may give you increasingly better approximations of
this value, but non-computable numbers can still be represented and
talked about by using symbols such as 3.1415... 0.999...

Then things got complex.

On Monday, 17 May 2021 at 08:15:50 UTC-4, timba...@gmail.com wrote:
> On Monday, May 17, 2021 at 7:30:01 AM UTC-4, Eram semper recta wrote:
> > On Monday, 17 May 2021 at 01:11:26 UTC-4, zelos...@gmail.com wrote:
> > > >Eudoxus of Cnidus used only **magnitudes** and that bullshit construction in the pdf you provided tries to build on his ideas albeit unsuccessfully.
> > >
> > > Who cares? We are not talking about his work.
> > >
> > > >Even the dumbest on this newsgroup know that you are an idiot. You were asked to demonstrate what that operator x (multiplication) means. Not only did you fail dismally in assuming that it means ac, but you failed to understand the question!
> > >
> > > There it is the multiplication of integers, duh.
> > Newsflash: Integers are RATIONAL NUMBERS.
> >
> > Eyes rolling and FACEPALM.
> This is a perfect instance of mathematical inversion of structure. Can it be stated that the rational numbers are constructed out of two integers?

NO! Because an integer is a RATIONAL NUMBER, so you immediately have circularity. Rational numbers came long before natural numbers or integers.

Study my article and learn, you dimwit!!!!!

https://drive.google.com/file/d/1hasWyQCZyRN3RkdvIB6bnGIVV2Rabz8w/

> Should it then be stated that integers can be constructed out of rational numbers?

This is how all numbers are constructed. If it is not a rational number, then you are NOT talking about NUMBER, but something else.

> Set theory ...

is a bunch of crap. PERIOD.

Eram semper recta <thenewcalculus@gmail.com> wrote:
> On Monday, 17 May 2021 at 08:15:50 UTC-4, timba...@gmail.com wrote:

[ .... ]

>> This is a perfect instance of mathematical inversion of structure. Can
>> it be stated that the rational numbers are constructed out of two
>> integers?

> NO! Because an integer is a RATIONAL NUMBER, so you immediately have
> circularity. Rational numbers came long before natural numbers or
> integers.

Rubbish. Integers are distinct from rational numbers, and have
properties (things like factorisation) that rational numbers lack.

Integers came before rational numbers. People learned to count before
they learned to divide numbers. A rational number is a "ratio", and
exactly what were these rational numbers ratios of? Integers, of course.

In modern mathematics, rational numbers are constructed from certain
ordered pairs of integers, so the integers come first.

>> Should it then be stated that integers can be constructed out of
>> rational numbers?

Definitely not.

> This is how all numbers are constructed. If it is not a rational
> number, then you are NOT talking about NUMBER, but something else.

That something else being a natural number, an integer, an algebraic
number, a transcendental number, a complex number, ....

>> Set theory ...

> is a bunch of crap. PERIOD.

Translation: Set theory is something not understood by John Gabriel.

--
Alan Mackenzie (Nuremberg, Germany).

On Monday, 17 May 2021 at 19:32:21 UTC+2, Alan Mackenzie wrote:
> Eram semper recta <thenewc...@gmail.com> wrote:
> > On Monday, 17 May 2021 at 08:15:50 UTC-4, timba...@gmail.com wrote:
> [ .... ]

> >> Should it then be stated that integers can be constructed out of
> >> rational numbers?

> Definitely not.

I agree with the overall sentiment, but that's not true: one can build numbers bottom up from the naturals, but it is also possible to go top-down from the reals.

By the way, the fact that theorems (to the extent that they apply) do not transpose easily if at all in standard mathematic, essentially due to a mixture of semantic with encoding aspects, is the main motivation for Univalent Foundations and the stronger notion of equality that it introduces ("equivalence is the same as equality"): such that e.g. the rational 2/1 is in fact the same as the integer 2 in that theory, and the same theorems apply to both.

Julio

Julio Di Egidio <julio@diegidio.name> wrote:
> On Monday, 17 May 2021 at 19:32:21 UTC+2, Alan Mackenzie wrote:
>> Eram semper recta <thenewc...@gmail.com> wrote:
>> > On Monday, 17 May 2021 at 08:15:50 UTC-4, timba...@gmail.com wrote:
>> [ .... ]

>> >> Should it then be stated that integers can be constructed out of
>> >> rational numbers?

>> Definitely not.

> I agree with the overall sentiment, but that's not true: one can build
> numbers bottom up from the naturals, but it is also possible to go
> top-down from the reals.

How, then, does one construct the reals in such a system? And having
the reals, how does one then derive the rationals?

> By the way, the fact that theorems (to the extent that they apply) do
> not transpose easily if at all in standard mathematic, essentially due
> to a mixture of semantic with encoding aspects, is the main motivation
> for Univalent Foundations and the stronger notion of equality that it
> introduces ("equivalence is the same as equality"): such that e.g. the
> rational 2/1 is in fact the same as the integer 2 in that theory, and
> the same theorems apply to both.

How can that be, when integers and ratonals both have properties that
the other doesn't? For example, the rationals are topologically dense,
the integers have factorisations.

> Julio

--
Alan Mackenzie (Nuremberg, Germany).

On Monday, May 17, 2021 at 11:39:54 AM UTC-7, Alan Mackenzie wrote:
> Julio Di Egidio <ju...@diegidio.name> wrote:
> > On Monday, 17 May 2021 at 19:32:21 UTC+2, Alan Mackenzie wrote:
> >> Eram semper recta <thenewc...@gmail.com> wrote:
> >> > On Monday, 17 May 2021 at 08:15:50 UTC-4, timba...@gmail.com wrote:
> >> [ .... ]
>
> >> >> Should it then be stated that integers can be constructed out of
> >> >> rational numbers?
>
> >> Definitely not.
>
> > I agree with the overall sentiment, but that's not true: one can build
> > numbers bottom up from the naturals, but it is also possible to go
> > top-down from the reals.
> How, then, does one construct the reals in such a system? And having
> the reals, how does one then derive the rationals?
> > By the way, the fact that theorems (to the extent that they apply) do
> > not transpose easily if at all in standard mathematic, essentially due
> > to a mixture of semantic with encoding aspects, is the main motivation
> > for Univalent Foundations and the stronger notion of equality that it
> > introduces ("equivalence is the same as equality"): such that e.g. the
> > rational 2/1 is in fact the same as the integer 2 in that theory, and
> > the same theorems apply to both.
> How can that be, when integers and ratonals both have properties that
> the other doesn't? For example, the rationals are topologically dense,
> the integers have factorisations.
>
> > Julio
> --
> Alan Mackenzie (Nuremberg, Germany).

all positive quantities are real
They are subject to a limited subtraction.

Mitchell Raemsch

On Monday, 17 May 2021 at 20:39:54 UTC+2, Alan Mackenzie wrote:
> Julio Di Egidio <ju...@diegidio.name> wrote:
> > On Monday, 17 May 2021 at 19:32:21 UTC+2, Alan Mackenzie wrote:
> >> Eram semper recta <thenewc...@gmail.com> wrote:
> >> > On Monday, 17 May 2021 at 08:15:50 UTC-4, timba...@gmail.com wrote:
> >> [ .... ]
> >> >> Should it then be stated that integers can be constructed out of
> >> >> rational numbers?
>
> >> Definitely not.
>
> > I agree with the overall sentiment, but that's not true: one can build
> > numbers bottom up from the naturals, but it is also possible to go
> > top-down from the reals.
>
> How, then, does one construct the reals in such a system? And having
> the reals, how does one then derive the rationals?

The same way one constructs the naturals in the other: postulate some primitives, in that case "continuity" in some form.

> > By the way, the fact that theorems (to the extent that they apply) do
> > not transpose easily if at all in standard mathematic, essentially due
> > to a mixture of semantic with encoding aspects, is the main motivation
> > for Univalent Foundations and the stronger notion of equality that it
> > introduces ("equivalence is the same as equality"): such that e.g. the
> > rational 2/1 is in fact the same as the integer 2 in that theory, and
> > the same theorems apply to both.
> How can that be, when integers and ratonals both have properties that
> the other doesn't? For example, the rationals are topologically dense,
> the integers have factorisations.

To the extent that they apply: i.e. there is an overlap as of course the integers are a subset as well as substructure of the rationals, and so on. In standard formal mathematics you can hardly even say that 2/1 the rational number is equal to 2 the integer.

Julio

On Monday, 17 May 2021 at 21:29:10 UTC+2, Julio Di Egidio wrote:
> On Monday, 17 May 2021 at 20:39:54 UTC+2, Alan Mackenzie wrote:
> > Julio Di Egidio <ju...@diegidio.name> wrote:
> > > On Monday, 17 May 2021 at 19:32:21 UTC+2, Alan Mackenzie wrote:
> > >> Eram semper recta <thenewc...@gmail.com> wrote:
> > >> > On Monday, 17 May 2021 at 08:15:50 UTC-4, timba...@gmail.com wrote:
> > >> [ .... ]
> > >> >> Should it then be stated that integers can be constructed out of
> > >> >> rational numbers?
> >
> > >> Definitely not.
> >
> > > I agree with the overall sentiment, but that's not true: one can build
> > > numbers bottom up from the naturals, but it is also possible to go
> > > top-down from the reals.
> >
> > How, then, does one construct the reals in such a system? And having
> > the reals, how does one then derive the rationals?
> The same way one constructs the naturals in the other: postulate some primitives, in that case "continuity" in some form.

.... then the rationals are some specific subset/substructure of the reals ....

Sorry, I do not have ready references but shouldn't be too hard to find.

> > > By the way, the fact that theorems (to the extent that they apply) do
> > > not transpose easily if at all in standard mathematic, essentially due
> > > to a mixture of semantic with encoding aspects, is the main motivation
> > > for Univalent Foundations and the stronger notion of equality that it
> > > introduces ("equivalence is the same as equality"): such that e.g. the
> > > rational 2/1 is in fact the same as the integer 2 in that theory, and
> > > the same theorems apply to both.
> > How can that be, when integers and ratonals both have properties that
> > the other doesn't? For example, the rationals are topologically dense,
> > the integers have factorisations.
> To the extent that they apply: i.e. there is an overlap as of course the integers are a subset as well as substructure of the rationals, and so on. In standard formal mathematics you can hardly even say that 2/1 the rational number is equal to 2 the integer.

Julio

On Monday, 17 May 2021 at 13:32:21 UTC-4, Shithead Scott Alan Mackenzie driveled:
> Eram semper recta <thenewc...@gmail.com> wrote:
> > On Monday, 17 May 2021 at 08:15:50 UTC-4, timba...@gmail.com wrote:
> [ .... ]
> >> This is a perfect instance of mathematical inversion of structure. Can
> >> it be stated that the rational numbers are constructed out of two
> >> integers?
>
> > NO! Because an integer is a RATIONAL NUMBER, so you immediately have
> > circularity. Rational numbers came long before natural numbers or
> > integers.

> Rubbish. Integers are distinct from rational numbers, and have
> properties (things like factorisation) that rational numbers lack.

Hello MORON. On occasion I will reveal your stupidity just for fun.
You are an imbecile who knows SHIT about mathematics.
How many times do I need to tell you that you are intellectually disabled?!

Integers ARE rational numbers, you fucking dumb Scot fool.

An integer p/q in particular, IS derived from a ratio of magnitudes p : q where p is composed of q and q is the chosen "unit magnitude".

Of course rational numbers have all sorts of properties, you infinite moron!!!!! There is nothing remarkable about this.

Get it piss head? LMAO.

>
> Integers came before rational numbers.

Bullshit. The way people counted before the Greeks was superficial shit which you find in set theory today. Little stones were used. Chuckle.

> People learned to count before they learned to divide numbers.

Bollocks! Till the operation of division was formally defined in geometry by my brilliant ancestors, no one knew how to divide rational numbers or even integers except for a very superficial understanding. Division with integers is very easy and in algebra nothing happens except perhaps the obelus operation.

> A rational number is a "ratio", and
> exactly what were these rational numbers ratios of? Integers, of course.

No moron. No. Book V of the Elements introduces the ratio concept long before Book 7 where number is introduced via the abstract unit.

Of course you are a fucking imbecile! LMAO.

>

> > This is how all numbers are constructed. If it is not a rational
> > number, then you are NOT talking about NUMBER, but something else.

> That so.. <snip-shit>

SHUT UP MORON. SHUT UP.

> Alan Mackenzie (Nuremberg, Germany).

On Monday, 17 May 2021 at 15:29:10 UTC-4, Fucking Italian MORON Julio de Egidio driveled:

> > > I agree with the overall sentiment, but that's not true: one can build
> > > numbers bottom up from the naturals, but it is also possible to go
> > > top-down from the reals.

Whether you agree or not you pathetic scrotum, it makes no difference. You are a NOBODY who knows shit.

> >
> > How, then, does one construct the reals in such a system? And having
> > the reals, how does one then derive the rationals?
> The same way one constructs the naturals in the other: postulate some primitives, in that case "continuity" in some form.

LMAO. Hey idiot. Shut the fuck up. You are a fool who knows shit. First try to understand what the word "incommensurable" means, you brain damaged Italian ape!!

> > > By the way, the fact that theorems (to the extent that they apply) do
> > > not transpose easily if at all in standard mathematic, essentially due
> > > to a mixture of semantic with encoding aspects, is the main motivation
> > > for Univalent Foundations and the stronger notion of equality that it
> > > introduces ("equivalence is the same as equality"): such that e.g. the
> > > rational 2/1 is in fact the same as the integer 2 in that theory, and
> > > the same theorems apply to both.

> > How can that be, when integers and ratonals both have properties that
> > the other doesn't? For example, the rationals are topologically dense,
> > the integers have factorisations.

Your head is topologically sparse. It's hilarious watching two baboons trying to discuss topics they have ZERO understanding about.

> In standard formal mathematics you can hardly even say that 2/1 the rational number is equal to 2 the integer.

You mean in "formal mythmatics"?

ROFLMAO.

"2/1 =/= 2" - Julio De Idioto.

>
> Julio

Julio Di Egidio <julio@diegidio.name> wrote:
> On Monday, 17 May 2021 at 21:29:10 UTC+2, Julio Di Egidio wrote:
>> On Monday, 17 May 2021 at 20:39:54 UTC+2, Alan Mackenzie wrote:
>> > Julio Di Egidio <ju...@diegidio.name> wrote:
>> > > On Monday, 17 May 2021 at 19:32:21 UTC+2, Alan Mackenzie wrote:
>> > >> Eram semper recta <thenewc...@gmail.com> wrote:
>> > >> > On Monday, 17 May 2021 at 08:15:50 UTC-4, timba...@gmail.com wrote:
>> > >> [ .... ]
>> > >> >> Should it then be stated that integers can be constructed out of
>> > >> >> rational numbers?

>> > >> Definitely not.

>> > > I agree with the overall sentiment, but that's not true: one can build
>> > > numbers bottom up from the naturals, but it is also possible to go
>> > > top-down from the reals.

>> > How, then, does one construct the reals in such a system? And having
>> > the reals, how does one then derive the rationals?

>> The same way one constructs the naturals in the other: postulate some
>> primitives, in that case "continuity" in some form.

> ... then the rationals are some specific subset/substructure of the
> reals ...

OK. That could be interesting, I've never seen it done that way before.
I suppose that's more or less the way Mr. Gabriel's construction of
numbers is supposed to work. Or something like that.

> Sorry, I do not have ready references but shouldn't be too hard to find.

Thanks, that's OK!

>> > > By the way, the fact that theorems (to the extent that they apply) do
>> > > not transpose easily if at all in standard mathematic, essentially due
>> > > to a mixture of semantic with encoding aspects, is the main motivation
>> > > for Univalent Foundations and the stronger notion of equality that it
>> > > introduces ("equivalence is the same as equality"): such that e.g. the
>> > > rational 2/1 is in fact the same as the integer 2 in that theory, and
>> > > the same theorems apply to both.

>> > How can that be, when integers and ratonals both have properties that
>> > the other doesn't? For example, the rationals are topologically dense,
>> > the integers have factorisations.

>> To the extent that they apply: i.e. there is an overlap as of course
>> the integers are a subset as well as substructure of the rationals,
>> and so on. In standard formal mathematics you can hardly even say that
>> 2/1 the rational number is equal to 2 the integer.

No, you would say there is a mapping of the integers into the rational
numbers, and then have to go through all the boring stuff about proving
the mapping preserves this, that and the other.

> Julio

--
Alan Mackenzie (Nuremberg, Germany).

On 5/17/2021 2:32 PM, Julio Di Egidio wrote:
> On Monday, 17 May 2021 at 21:29:10 UTC+2, Julio Di Egidio wrote:
>> On Monday, 17 May 2021 at 20:39:54 UTC+2, Alan Mackenzie wrote:
>>> Julio Di Egidio <ju...@diegidio.name> wrote:
>>>> On Monday, 17 May 2021 at 19:32:21 UTC+2, Alan Mackenzie wrote:
>>>>> Eram semper recta <thenewc...@gmail.com> wrote:
>>>>>> On Monday, 17 May 2021 at 08:15:50 UTC-4, timba...@gmail.com wrote:
>>>>> [ .... ]
>>>>>>> Should it then be stated that integers can be constructed out of
>>>>>>> rational numbers?
>>>
>>>>> Definitely not.
>>>
>>>> I agree with the overall sentiment, but that's not true: one can build
>>>> numbers bottom up from the naturals, but it is also possible to go
>>>> top-down from the reals.
>>>
>>> How, then, does one construct the reals in such a system? And having
>>> the reals, how does one then derive the rationals?
>> The same way one constructs the naturals in the other: postulate some primitives, in that case "continuity" in some form.
>
> ... then the rationals are some specific subset/substructure of the reals ...

start with natural numbers

>
> Sorry, I do not have ready references but shouldn't be too hard to find.
>
>>>> By the way, the fact that theorems (to the extent that they apply) do
>>>> not transpose easily if at all in standard mathematic, essentially due
>>>> to a mixture of semantic with encoding aspects, is the main motivation
>>>> for Univalent Foundations and the stronger notion of equality that it
>>>> introduces ("equivalence is the same as equality"): such that e.g. the
>>>> rational 2/1 is in fact the same as the integer 2 in that theory, and
>>>> the same theorems apply to both.
>>> How can that be, when integers and ratonals both have properties that
>>> the other doesn't? For example, the rationals are topologically dense,
>>> the integers have factorisations.
>> To the extent that they apply: i.e. there is an overlap as of course the integers are a subset as well as substructure of the rationals, and so on. In standard formal mathematics you can hardly even say that 2/1 the rational number is equal to 2 the integer.
>
> Julio
>

Alan Mackenzie presented the following explanation :
> Julio Di Egidio <julio@diegidio.name> wrote:
>> On Monday, 17 May 2021 at 21:29:10 UTC+2, Julio Di Egidio wrote:
>>> On Monday, 17 May 2021 at 20:39:54 UTC+2, Alan Mackenzie wrote:
>>>> Julio Di Egidio <ju...@diegidio.name> wrote:
>>>>> On Monday, 17 May 2021 at 19:32:21 UTC+2, Alan Mackenzie wrote:
>>>>>> Eram semper recta <thenewc...@gmail.com> wrote:
>>>>>>> On Monday, 17 May 2021 at 08:15:50 UTC-4, timba...@gmail.com wrote: [
>>>>>>> .... ]
>>>>>>>> Should it then be stated that integers can be constructed out of
>>>>>>>> rational numbers?
>
>>>>>> Definitely not.
>
>>>>> I agree with the overall sentiment, but that's not true: one can build
>>>>> numbers bottom up from the naturals, but it is also possible to go
>>>>> top-down from the reals.
>
>>>> How, then, does one construct the reals in such a system? And having
>>>> the reals, how does one then derive the rationals?
>
>>> The same way one constructs the naturals in the other: postulate some
>>> primitives, in that case "continuity" in some form.
>
>> ... then the rationals are some specific subset/substructure of the
>> reals ...
>
> OK. That could be interesting, I've never seen it done that way before.
> I suppose that's more or less the way Mr. Gabriel's construction of
> numbers is supposed to work. Or something like that.
>
>> Sorry, I do not have ready references but shouldn't be too hard to find.
>
> Thanks, that's OK!
>
>>>>> By the way, the fact that theorems (to the extent that they apply) do
>>>>> not transpose easily if at all in standard mathematic, essentially due
>>>>> to a mixture of semantic with encoding aspects, is the main motivation
>>>>> for Univalent Foundations and the stronger notion of equality that it
>>>>> introduces ("equivalence is the same as equality"): such that e.g. the
>>>>> rational 2/1 is in fact the same as the integer 2 in that theory, and
>>>>> the same theorems apply to both.
>
>>>> How can that be, when integers and ratonals both have properties that
>>>> the other doesn't? For example, the rationals are topologically dense,
>>>> the integers have factorisations.
>
>>> To the extent that they apply: i.e. there is an overlap as of course
>>> the integers are a subset as well as substructure of the rationals,
>>> and so on. In standard formal mathematics you can hardly even say that
>>> 2/1 the rational number is equal to 2 the integer.
>
> No, you would say there is a mapping of the integers into the rational
> numbers, and then have to go through all the boring stuff about proving
> the mapping preserves this, that and the other.

Yeah, and for Q you also need equivalence classes because you generate
too many symbols having the same value.