Today in my foundations class the professor constructed the integers. After class I asked him about who came up with all that, because last time he told us exactly who 'invented' the naturals, but he didn't know.
I googled it and I found some names of people in the 19th century who were interested in constructing the integers but not the name of the one guy who did this construction.
By the way, I am talking of the construction where integers are the equivalence classes of the set of naturals under the relation (a,b)R(c,d) iff a+d=b+c
Does anyone here know who invented this? Veeky Forums is my last resort. I really want to know.
>last time he told us exactly who 'invented' the naturals
What the fuck? Natural numbers have literally been known throughout the human history. I bet fucking cavemen were already like "one stick plus one stick is two sticks".
Nicholas Torres
The axioms of arithmethic that give rise to the naturals were made by Peano.
John Williams
No. The axioms were used a lot earlier than Peano had even born. At most he might have been the first one to formally write them down. Egyptians worked with natural numbers literally three thousand years ago.
Grayson Scott
No, no one used axioms before. Arithmethic and algebra used to be in the air, or supported by geometry as was the case with the greeks.
Peano invented the axioms which are really abstract so I doubt a 3000 bc egyptian could even comprehend them.
Lucas Harris
you really think humans evolved higher intelligence in just a few thousand years
Owen Perry
Abstract mathematics (formal, rigorous and axiomatic) needs much more than just intelligence. Motivation and a culture was also needed.
Google this shit. Up to the 17th century people did not believe in negatives and before the 19th we did not even know how they 'looked inside'.
Luke Murphy
Hell, probably. Considering the leap that was made in a few hundred thousand years to the neolithic era. Its not a huge jump to consider it.
Easton Fisher
Peano axiomatized the naturals, yes. One way to construct the integers is to notice that the naturals under addition satisfy all but one group axiom, they lack the non-zero inverses. To have the set (/group) of integers is to have a negative version of each positive natural added to the set of natural numbers, but this is to have the additive group generated by naturals, and this is a way to construct the integers. More precisely, for any natural number k, define -k to be such that k+(-k)=-k+k=0, and then define [math]\mathbb{Z}= \{ k, -k \ | \ k \in \mathbb{N} \}[/math].
We have 0 as the smallest natural here.
Joshua Young
Your construction needs the foundation of algebra.
The construction I speak of only needs that the naturals exist and is therefore more elementary and probably better and more meaningful.
Jeremiah Wilson
In case people are having a hard time, the construction I speak of is this one
But like every other article explaining it, it speaks nothing of who actually came up with it. Why? This shit is top tier mathematical beauty. Why is the person who invented it not a math meme like Ramanujan?
Henry Green
>before the 19th we did not even know how they 'looked inside'. wat
Nolan Carter
I am talking about the way we construct them, what they are as sets.
The natural numbers of the form 1, 2, 3, etc. are just their symbols. Inside they look like {{},{{}}}
I think that's 2.
For the integers again 0 , -1, 3, etc. are just symbols.
Inside the number zero looks like [0,0] and that looks like {(0,0),(1,1),(2,2),...}
Then you could also make it look weirder by applying the set definition of ordered pair and then changing the numbers of the sets that represent them. You will have a fucking mess though but that is how those numbers look and implicitly when doing arithmethic we are messing with that.
Ian Fisher
thats not 'inside'.
if anything its to the side since it uses alternative symbols.
Asher Hernandez
That is inside.
In modern mathematics everything is supposed to be a set. When you do operations you are moving sets and all the sets we know are constructed with the sets generated by the axiom of infinity.
Owen Ramirez
This guy is right.
You're retarded.
Logan Campbell
kek, im not the guy thinking numbers have an inside
Caleb Hernandez
1. Peano, the short answer is Peano. See the rest for a longer answer.
2. Mathematics is not invented, but discovered. But this is a philosophical point where a simple majority of Veeky Forums incorrectly disagrees with the above/me, because they have not thought the matter all the way through. It's not directly pertinent to your question OP, but words mean things, and you should at least be aware that there are people who honestly hold either opinion.
3. Actually, the philosophical inquiry is very important to the language that you've used in your OP, OP. Many anons have read your OP and come away with the idea that "inventing integers" is absurd (they're right - it is). Ironically, their intuition on this point is correct, and their responses buttress my assertion that math is a discovery, and not an invention. The Egyptians figured out that there are these things called natural numbers, fractions, and units of measurement (even if they didn't conceive of such in modern terms, they had a good understanding of the rules of arithmetic, fraction manipulation, and unit conversion). But of course the Egyptians did not invent natural numbers, any more than Peano did. Rather, they simply discovered them, and what they did invent, did contrive, was a way of talking about them and manipulating them which was in accord with what they knew perfectly well natural numbers are, while at the same time meeting their respective goals for how to speak and write of numbers.
4. But it's clear from reading the title of the OP, that the OP also knows that no one "invented" numbers: what he's asking is, "who came up with the modern convention for speaking of numbers that I saw in my class?" To this, the most likely answer would seem to be Peano (though I don't know that for a fact). Your googling sounds like you were looking in the right place. For very closely related discourse about natural numbers, see Russell's Introduction to Mathematical Philosophy, which builds on Peano.
Chase Smith
>Mathematics is not invented, but discovered your a fucking idiot,
axioms are invented, theorems based on them are then discovered. What you are talking about is the mapping between numbers and real world objects, that too, is invented based on properties that are discovered about the objects. We invent axioms such that the results that are discovered can easily be mapped to the discovered properties of objects using maps we invent.
Adrian Bennett
>Peano, the short answer is Peano. See the rest for a longer answer.
No, he did the axioms for the natural numbers which are used again but he clearly did not make the integers. The construction is so elaborate that I doubt he did it.
Also when I investigated I found that people only cared about this in the 19th century so this particular construction probably came about in the 19th century, not in Peano's time.
Nathaniel King
Clever boy. Now ask yourself why it is that certain particular axioms are adopted, during a creative process of doing math. This entails re-reading your latter sentence. Notice that word that you used twice in your latter sentence. No, this isn't trickery, or misunderstanding. It's important.
It's almost as if certain axioms intrude, force themselves upon us, in accord with what is the case. Because it is. Prime numbers, say, will go on being prime long after every human is dead, waiting to be discovered agian by species 8472, elsewhere.
Joseph Garcia
It's like you really believe in the World of Forms.
Brayden Bell
You're wrong, most of peano's axioms (axioms 1 to 8) are just self evident facts about counting objects, the assumption that 1 is the smallest number, and the assertion that you can make more numbers by adding 1 repeatedly, which every person throughout history has understood. The only axiom that wasn't commonly used was axiom 9, the induction axiom, but it isn't necessary to create the natural numbers, as presburger arithmetic creates the set of natural numbers with no concept of mathematical induction.
Jeremiah Smith
Kronecker
Christopher Walker
>are just self evident facts about counting objects
Axioms are usually defined as self evident facts. The thing is that they were never laid out before Peano and that was the game changer.
>but it isn't necessary to create the natural numbers
You are technically right but there is no point in having natural numbers without the properties that the axiom of induction gives it.
Do you know how to prove that addition is commutative? Axiom of induction. Multiplication is associative? Axiom of induction. Prove that a + x = b + x implies a = b? Axiom of induction.
Without the axiom of induction the natural numbers are merely a list with no structure, properties or even meaning.
Source? I checked his wikipedia page and even though he is listed as someone who believed in constructing the integers, there is no reference that says that he was the one who came up with this specific construction.
I need answers. Please.
Owen Watson
Where mathematical objects are concerned, yes, that's about right. Moreover, anyone can arrive at about the same position with a little clear thought. A certain irony in this is that I am not religious. Please remove your trip btw.
Notice that although Peano's formulation is one of the better-known ones (which is why I alongside other anons have mentioned it repeatedly ITT), these two anons are quarreling over "authorship" of a system that gives rise to the naturals. Hm, it's almost as if the topics under discussion are exterior to humans, something not really made by them but intelligible to them, constantly waiting to be found or re-found, at any period of history. What do you call it when there's something out-there, in-the-world, that humans don't really /make/ as such, but /find/?
I do see now that the OP has posted a helpful link . OP, I have been flipping through some of my Russell books and skim/comparing them to this and the OP, but I still haven't found any definite comparison. IMP contains one relation-passage about integers themselves however. Compare pp. 112-3 (that is, "p.64"):
Got interested and saw that he didn't even define multiplication, just addition and equality.
>Presburger arithmetic is much weaker than Peano arithmetic
Source: wikipedia
Again, what is the point of having a mere list of numbers. There is really no point in making numbers if we can't make good algebraic structures out of them. A theory of arithmethic without multiplication is a theory of arithmethic I do not care for.
Carson Gonzalez
They worked with the natural numbers but it doesn't mean they were understood in the way it is now using Peano's axioms. It's like how Newton/Liebniz worked with calculus but the axioms and theorems weren't developed until later.
Camden Gonzalez
Bump.
There has to be at least one guy here who is a huge math history buff.
Jason Jones
I actually am, and I've made some effort to answer OP's question, but then I'm not a PhD either. I still haven't exhausted the Russell possibility, however.
Ian Thomas
>Without the axiom of induction, the natural numbers are merely a list with no structure, properties, or even meaning So? You still can use presburger arithmetic to count things, which is the defining property of the natural numbers, induction and multiplication are just add-on components, like how the complex numbers are add-on components for the real numbers.
Jonathan Wood
>i can't multiply the numbers together, so that means they aren't natural numbers natural numbers only need to count things, multiplication is just an extra action
Easton Perez
>he natural numbers of the form 1, 2, 3, etc. are just their symbols. Inside they look like {{},{{}}}
That's a construction of natural numbers using axioms of set theory. The set notation is also just symbols. It's the numbers' "true "form".
Dominic Smith
*it's NOT the numbers' "true form"
Christian White
OP, the more I skim through IMP, the more convinced I am that it's among the closest items that you will find to answer your question. Check my citation earlier in the thread and read down just a bit. You will see Russell mention a technique for dealing with division by simply getting rid of it for a moment via classes, multiplication etc in exactly the same sort of way that your link gets round subtraction for a moment by doing its thing with ordered pairs, classes, and division (and never speaking of subtraction). The relevant passage is just below the one that I had pointed out above.
Moreover, in the course of same, Russell makes clear that the lower-case omega notation to denote the whole numbers, which is repeated throughout your link, is also due to Cantor:
"We start with the series 1,2,3,4... n ..., which, as we have already seen, represents the smallest of infinite serial numbers, the sort that Cantor calls [math] \omega [/math] ." -Bertrand Russell, Introduction to Mathematical Philosophy, Opening passage of Chapter nine (skip down a few paragraphs).
My point that this is still just a slight extension of Peano (which remains where this conversation really starts, to the historical narrative of the thread) still stands. "The short answer is Peano" --- by way of Russell perhaps, and/or Cantor.
Ian Sanchez
for my second "division" in this post, instead read "addition". a gaffe.
Parker Evans
Nomar Integerus
James Thomas
>a method of constructing the integers is "top-tier mathematical beauty" somebody shoot me
Sebastian Thompson
This is something that I had had in the back of my head, but OP has been so nice and persistent that I didn't want to bring it up, because I do enjoy discussing the history of math. Now the door has been opened and let's go for it.
OP's thing has been "this exact construction/problem/technique was done. Where does it come from, historically?" He has been very, very insistent on finding an answer, and intelligent and civil about doing so, to his credit. But mathematically speaking, the technique itself isn't all that amazing --- /except from a historical point of view/, and not a mathematical one.
On the other hand, let's be more charitable to the OP. It sounds like the OP is an undergrad who is branching out, is "catching on" and is liking what he finds, what he has to think about. Good for him! In this sense, your "somebody shoot me" is wholly invalid. I can remember first appreciating certain "simple" arguments as what is instead a really rather elaborate reasoning process that we take for granted, once we have internalized them. It's the joy of discovery, of knowledge creation. And it's non-trivial.
Still, OP's specific instance is rather boring. OP, let me suggest to you that the type of thing that you have before you may have been considered by multiple mathematicians as a conceptual exercise, which they eventually found to be straightforward, and so they didn't "publish a great paper" about same, but instead subsumed the suggestion into their larger project. Perhaps the reason why you can't find this exact development in the history of math is because it's one of those things that were "left as exercises", as it were. I therefore refer you back again to IMP as a closest available historical comparison.
Jaxson Harris
OP here. I appreciate your sources and you are probably right, I just thought someone would have their name on it.
Regarding the beauty of it, I don't know man, it looks really clever to me. By the way, the foundation class I'm taking is a freshman course and I am an undergrad.
Anyways, this isn't exactly the same as the Peano axioms. The Peano axioms give a first-order characterization of the natural numbers (which isn't categorical btw). The Grothendieck group says that if you have ordered pairs and naturals, you also have integers, so it's more like an encoding or a construction. We don't really "construct" the naturals, you have to basically assume they exist in one form or another, and the Peano axioms aren't even really the best way to do that.
Carter Nguyen
We are at least in agreement that, as far we we two can tell, OP's closest bet is Cantor/Russell.
Jack Torres
>the universe invented a human mathematical abstraction Well I guess you're technically right...
Jordan Reed
This definition is essentially equivalent to the modern one. He first identifies the natural number n with the function (relation) f(x) = x + n, and then defines -n to be its relational converse, and so [math]\mathbb{Z}[/math] is the set of all these relations. If you define a relation as a set of ordered pairs then this is exactly the same definition.
That book is supposed to be a less technical explanation of the ideas in Principia Mathematica, and it seems Z was constructed in the same way there:
"* Here the type "*w" is the type of converses of relations of type w, i.e. the type of the negative integers in order of magnitude, ending with - 1, « being the type of the positive integers in order of magnitude, and therefore *w+« being the type of negative and positive integers in order of magnitude. "
Joshua Lopez
>then defines -n to be its relational converse
I don't like this. This definition pre supposes the existence of algebra and the idea that inverses are even a thing.
Think about it. Defining -1 to be the number such that 1 + (-1) = 0 is no different from defining i as the square root of -1. This kind of thing should only be used when studying algebra itself. A theory of arithmethic should be the most fundamental part of math (after set theory) and from arithmethic we should define algebra as its generalization.
I think this is historically intuitive given that arithmethic came before algebra so if you define arithmethic in terms of algebra you are pretty much ignoring mathematical history, and lets not forget that you are basically trying to prove that the square root of 2 is irrational by saying that the contrary contradicts Fermat's last theorem. It is just not necessary to do it that way and it is way overkill.
The construction being discussed here needs nothing other than set theory and natural numbers while this construction needs set theory, natural numbers AND algebra.
Samuel Brooks
It doesn't need algebra. All it uses is relations, which are just as set-theoretic (aka logical) as equivalence classes.
>I think this is historically intuitive given that arithmethic came before algebra so if you define arithmethic in terms of algebra you are pretty much ignoring mathematical history
This is idiotic: Russell's definition is in fact MORE INTUITIVE and is probably the original one so gtfo with historical arguments.
It's more intuitive because positive integers can literally be identified with translations to the "right", and then obviously negative integers are just translations to the "left." Every kid learns about the number line this way, so why not make it the definition?
Plus, algebra is cooler than logic.
Thomas Morales
I did try skimming through PM to see if I could see same to OP's question, but y'know, lots of noise.
Sebastian Thompson
Turns out Frege used the same concept in his Grundgesetze. Amazingly the idea actually goes back to Gauss, picrelated. He probably didn't think of it as a way to construct numbers, but instead as a way of interpreting them.
