# Video: How do we know that 1+1=2? A journey into the foundations of math.

AskAMathematician.com presents a lecture on the foundations of math and whether we really can know that one plus one equals two. How was math invented? Where does mathematics come from? Are the axioms of math provable? Is math true? Can it be proven on purely logical grounds? Can it be demonstrated empirically? Can it only be justified from a pragmatic perspective? These are some of the questions discussed in the three videos below.

Part 1 of 3:

Part 2 of 3:

Part 3 of 3:

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### 18 Responses to Video: How do we know that 1+1=2? A journey into the foundations of math.

1. Sharkey says:

Great video.

I recently got into a discussion on how the universe’s exclusion principle as described by Pauli is a foundation for mathematics: since electrons, protons and neutrons exhibit space-occupying behaviour, the common elements in our experience follow the axioms of set-theory (at least in the finite cases), and so math is applicable to the real-world. If there was no exclusion principle, then our math would be much different (assuming we were here to derive math at all).

2. Mathematician, I really enjoyed these. Surely there are more to come?

I do feel though that cramming so many topics together seems overly ambitious, as the connections between them feels weak.

The audio track needs to be normalized, by the way, the volume is kinda low.

3. DFB says:

For some reason I was under the impression that Bertrand Russell had taken care of the 1+1=2 problem. Wasn’t there three-hundred and something pages in the Principia Mathematica that proved it once and for all using set theory without the axioms? What else have they been lying to me about?

Okay, well Wikipedia says that he used the axioms of type theory plus three other axioms. I guess I was misinformed. Bah!

What’s the difference between “set theory” and “type theory?” It seems like you alluded to it in the video.

4. Sharkey says:

DFB: Set theory (or naive set theory) is vulnerable to paradoxes; the best known is Russell’s Paradox, the set of all sets that don’t contain themselves. Type theory was an attempt to remove these paradoxes by imposing a class hierarchy: sets could only contain objects of a “lesser” class.

The hope was that type theory would provide a solid (non-paradoxical) foundation for mathematics. That hope was dashed by Godel’s incompleteness results; however, type theory found another use in modern computer languages, so not all was lost…

5. Francis says:

I greatly enjoyed your talk 🙂 You know, I wonder if you could adapt this to a TED talk, I’m sure that audience would find this fascinating as well.

One old philosophical question that your talk reminded me of is the age old question, “Is mathematics discovered, or invented”? I personally believe that the answer is somewhere in between these two choices. In some sense, mathematics describes characteristics and behaviours inherent in the universe, so you could think of it as a “discovery”. But in another sense, it describes things that have no relationship with the natural world, it can be a human “invention”. I’d like to hear your take on it 🙂

I have heard of Godel’s incompleteness theorem before, but I had not heard this idea that math cannot answer “Is there a cardinal number between aleph-0 and aleph-1?” I’d love to hear more examples and descriptions of where the limit between what math can prove and express, and what it cannot.

6. Questioneer says:

I saw your video about how we know 1+1=2. Anyway, my philosophy on the subject is that mathematics is ultimately a system of tautologous statements. For instance, we know that one plus one equals two because two is defined as one and one, so the statement boils down to “one and one is the same as one and one”.

Believing such a statement only requires that we know that can believe tautologies. How can we know that A = A, that B = B, that one and one = one and one? My answer: because we directly experience the truth of these statements. By direct experience we can determine that things are the same as themselves. And that type of direct experience cannot be doubted, therefore it needs no defense.
What do you think?

7. Mathematician says:

If math is true only in the same sense that any set of tautologies is true, then it is a very trivial and uninteresting form of “truth”. In that case, “1+1=2″ is just a complicated way of saying “A is A.” I think that many people who claim that math is true mean something more than this. However, if you take the axioms as given, then it is indeed true that everything else follows just by definition and by application of the axioms.

8. Sounds to me that you two’s thoughts agree, that the axioms are devised & modeled on what we intuitively think of as true, and what follows are then logically (tautologically?) true. My wording is probably not very good though.

I’m sure I’ve heard many times in class when professors teach a new definition/theorem, it goes like this: “So we want to define XXX, and intuitively it should do YYY and ZZZ, so let’s assume YYY & ZZZ… It turns out that this works perfectly/is just what we want/etc. except…”

9. EloquentMath says:

Obviously I’m wrong here, seeing as I have only a GCSE certificate in Maths, not a University degree, but I can’t quite comprehend that the positive integers are the same size as the integers. Say the positive integers are given the value of ∞, wouldn’t all the integers simply be 2∞+1? The “+1” being 0 and the 2∞ being because there is the same infinity of negatives and positives so if you add them together it would be 2∞? I’m assuming I’m vastly wrong and making a lot of assumptions but well, I have to ask.

10. The Mathematician says:

Hey EloquentMath,
Here is one way to think about it: Take the positive integers, and start renaming them. Rename
1 as 0,
2 as 1,
3 as -1,
4 as 2,
5 as -2,
6 as 3,
7 as -3,
8 as 4,
9 as -4,
and so on for all of them. Once this renaming has been completed for all of the positive integers, we see that now they just become the entire set of integers! But the size of a set doesn’t change simply be renaming the things in that set. So since we can rename the elements of the positive integers to turn it into the integers, the set of integers can’t be larger than the set of positive integers!

11. EloquentMath says:

Yeah, I understood that in the video and that makes a certain degreee of sense. But, if say infinity was defined to ‘x’, the positive integers would go 1, 2, 3… x right? But all integers would go -x… -3, -2, -1, 0, 1, 2, 3… x correct? So I just can’t see them being the same size.

12. Paul says:

“Two times infinity” doesn’t really make sense, nor does something like “infinity plus 1.” I think there’s a post on infinity here somewhere…

Wanting to define infinity as “x” seems to suggest that you still think of “infinity” as the same kind of gadget as a finite number, but unfortunately it isn’t.

13. EloquentMath says:

Of course I can understand that infinity is not a finite number, but suggesting one infinity is larger than the other also means that it has been thought of as a regular number. I can understand the concept but just struggling to see the actual concrete evidence.

14. Paul says:

Not really… rarely is infinity thought of like a regular number, and certainly not in this case. Infinity is a concept that our common sense of numbers and quantities don’t apply very well to.

If you go back to the second part of the video, you’ll see the Mathematician defining a way to compare the size of finite sets, then apply it to infinite sets. This is about the point where our intuition stops working, and a subset that is “missing elements” can now have the same “infinite size” as the whole set.

The Mathematician’s response 4 posts above, which he also said in the video, is in fact a real proof in casual wording, so that is as good a concrete evidence as you’ll get.

I hope the hosts here don’t mind me pointing to outside references, but the video by “Sixty Symbols” on infinity is worth checking out.

15. The Physicist says:

S’cool.

16. EloquentMath says:

Thanks I’ll check it out, sorry to seem stupid and I can see the logic but just I guess am used to dealing with finite numbers so need to break that cycle I guess.

17. The Physicist says:

Infinity is weird stuff.
There’s a post in the works.

18. Luther Hall says:

first i am not arguing but i am arguing a point and am open minded to your opinion.
1+1=2 can be proven !
1 does not just describe quantity. for example 1 chicken and 1 chair is in a fenced area and the question is ask “how many in there” the answer is 1, 1, or 2. but the first question back at you is “how many of what”(unless you were killing chickens , or moving chairs, or grouping items) So numbers describe quantity and type. its perpendicular description of sorts.
but lets go to gel balls. 1gb + 1gb = ?? you say its 1 or 2 . I say wrong based on my vector of perpendicular description. Lets see. 1 blue gb + 1 yellow gb = either 2 gb balls
or a completely entity which was not in the equation 1 green gb. another eg: 1gb (small) + 1 gb (medium) = 2gb or new entity(if they merge ofcourse) 1 large(er) gb.
What say you.