I've noticed a few word choices or errors that makes the statements confusing or incorrect so I'll put them here: 4:26 When I say #2 is false for 0, it would mean 0 has no successor (It does: it's 1). Rather, I should have phrased it that no number's successor is 0. I hope this is clear with the explanations thereafter. 5:38 -successor- predecessor 6:19 It should have said that if 0 has some property, and if a having with that property implies S(a) also has that property, all natural numbers have that property. Sorry for these errors I hope you don't get confused by it. If you are here as part of the SoME1 peer review, thank you so much for taking your time to watch this video. If anyone is really low on time, you can skip to 8:58 or 11:11 where I prove, and give a conclusion on the video! Thank you so much for watching!
I believe your correction of the fifth axiom is still inexact. You didn't quantify "a" in the first place, so it's really not clear what it means. It could either mean: - for every a in N, if 0, a, s(a) share a property, then all natural numbers have that same property. That's not the fifth axiom however. Moreover, it's false in natural numbers. For instance: 0, 1, s(1), all have the property of being less than 3. Clearly not all natural numbers are less than 3. or it could mean: - if 0 has a property, and for every a in N, a and s(a) have that property, then all natural numbers have that property. At this point it just becomes equivalent to "if, for every a in N, a has a property, then all natural numbers have that same property", which clearly is of little use. The correct phrasing should be something like: if 0 has a property, and if it's true that the same property belongs to s(a) if it belongs to some a, then all natural numbers share that property
@@betterpkm i think the clearest restatement would be to change the order of everything: 'given some property such that, some a having it requires that its s(a) has it, then if 0 has that property, therefore it is one which all natural numbers share.' (or, less concisely: 'given some property such that, some natural number a having it requires that its immediate successor i.e. s(a) has it, which, consequently all further successors will have it, then if 0 i.e. the initial natural number has that property, therefore it is a property which all natural numbers share since all natural numbers besides 0 ultimately succeed from 0.')
The lecture: Proof for 1+1=2 The homework: Prove that 7+13=20 The brain teaser in the book: Prove that 3-2=1 The test: Prove that 3*4=12 *The exam: Prove the Collatz conjecture using only successors, 0 and 1.*
Thanks mate my child had some trouble keeping up in their maths classes, this video has helped them a lot. Could you make another video proving that 1+2 = 3 soon?
This whole proof reminds me of freshman year in school with some mind-altering substances, and suddenly wondering "What if C-A-T really spelled dog???" "1+1 = 2 because 1 = 0+1 and 0+1+1 = 2. But if we define the successor to 0 as 一 then we'd say 0 + 一 = 一 and 0 + 一 + 一= 二 !!"
I always thought this was one of those topics that required either 2 seconds or +10 hours of discuss with no inbetweens, this video is almost a miracle.
Math is more enjoyable than Physics tho, even tho I never used most of the things that I learned on my life (or to be exact I don't even know what they are used for)
I always wondered how you could prove that 1 + 1 = 2 because that in of itself always seemed so fundamental to me. I mean, like you said, it's the first thing ever taught when kids are first learning math. I could not comprehend how the hell you could get any more basic until I watched this video. Thank you
this first symbol is called "one" and represents loneliness the second symbol is called "plus" and represents the end of loneliness the third symbol is called "one" and represents intrusion, the fourth symbol is called "equal" and represents change , the fifth symbol is called "two" and represents the mistake since 1+1=5
@@wosandakeweenjayaweera10-h48 if lets say were dealing with a massive number like 10^100, or larger, all numbers can essentially seem like 0 to that number, and hence you can say they are approximately each other in reference to said large number
@@o_enamuelThere are always cool basic things that make me rethink, like 5-2=3, but counting from 2,3,4,5 gives 4 numbers. (This was an important realization for off-by-one errors in programming) And 100^2 is 10000, but any number less than 1, squared, becomes exponentially smaller, like 0.5^2=0.25
You can use the 3 axioms of Peano, where the inductive axiom (technically axiom scheme) has "properties" instead of "subsets", and a hidden 0th axiom that governs the use of "=" and basically tells us that "successor" is a function: 0=0 Axiom scheme: m=n => S(m)=S(n), where m,n are natural number variables Together with the 3rd axiom, this proves that any symbol for natural numbers is equal to itself; and I think that, more generally, all three properties of "=" are now theorems, using again the 3rd axiom. Maybe. Yes, I tried doing it and I think I need axioms for "replacing formulas" now, like in λ-calculus.
1 plus 1 is 10 because I used the binary system instead of the decimal system that we normally use. 10 has the first digit marked as 1 so we need to add 2 because the binary system multiplies it self by 2 every digit from right to left. 1001001 is 73 because the last digit is marked as 1 indicating that it is an odd number. 1 plus 8 plus 64 is 73. Search this topic for more information.
i mean, that doesn't change anything since 2 and 10 are two ways to represent the same quantity using different symbols. we can say that 3,1415926... = π because "π" is a way to represent the value and not the value itself.
For the longest time I believed that 1+1=11. But now I’m convinced that I’m wrong, and it is 2. Time to say sorry to my 2 year old cousin that I taught him the wrong equations
I think you got the Peano axioms for natural numbers wrong. You say that: 1. 0 exists 2. For every natural number a, the successor of a, S(a), exists 3. Axiom 2 is false for 0. But that's a contradiction! I think you meant that: 3. 0 is the successor of no natural number Also, I'm not sure that axiom 5 defines the natural numbers. You could have the property "a is even" (whatever that means at that point), which contradicts your axiom, because S(a) would be odd for every natural number. The Peano axiom is: 5. If 0 is in a set A, and if for every natural number a in A, S(a) is in A, then A = N I think you tried to simplify this axiom, but I don't think it can :/
Isn't your 5 an embedding of N in A? By your definition you could have elements in A that are not natural numbers and you would still incorrectly conclude that A = N, when N is a subset of A.
@@tricky778 Indeed, technically it is. However, this embedding is simply a bijection (actually the identity) from N to A by the axiom. It defines N as the only set of numbers that follow axioms 1-4. It is not a conclusion, it is a definition Assume there was a number a in A that could not be reached by successive applications of S from 0. Then you could construct a strictly decreasing sequence starting from a that never reaches 0, which we know is impossible in N.
@@amarasa2567 oh, that wasn't an alternative to the collection of the other axioms but an additional one? Or 5 is missing the statement that 'forall a in A st. a =\= 0, exists b in A st. a = S b'?
@@tricky778 that looks like an equivalent formulation, though it arises from slightly less elementary notions (you basically used the basically the predecessor)
So it is an agreement that addition is a movement in a defined sequence of some symbols (numbers). Part of the agreement is that "+1" operator returns directly the next number in the sequence. So if we have "standardized" sequence of natural numbers: 1, 2, 3, 4, 5, ... that means 1+1=2. So in a nutshell it is an agreement about meaning of those symbols ( = and + included).
[Disclaimer: I'm quoting myself from a longer comment I made elsewhere. Not that I love repeating myself, but rather, your comment seemed to imply that Whitehead and Russel were unnecessarily verbose in their proof. So I thought I'd set that straight.] Well, to be fair to Whitehead and Russell, before proving 1+1=2, they had to rebuild the foundations of math, since Russell-with his eponymous paradox-had just blasted Cantor's 'naïve' set theory to smithereens. As Carl Sagan said, *_“If you wish to make an apple pie from scratch, you must first invent the universe.”_* And that's what W&R did in their 3-volume _Principia_ (1910-1913). Ironically, proving 1+1=2 is fairly trivial using the axioms of _Peano Arithmetic_ (ca. 1889), as this video shows. So when W&R proved the same thing within their reformulated set theory, what they were really doing is showing that it is consistent with arithmetic; that there are no nasty surprises of the kind that destroyed Cantor's set theory.
@@synchronium24 yea I'm not an expert myself but that's the story as I understand it. Russell was trying to make sure that self reference was impossible in this system to prevent what happened to naive set theory, but godel showed that you can force self reference on any system that's sufficiently powerful (arithmetic being one); the problem being that with self reference you can create the liars paradox.
Even easier: 1) Use roman numerals 2) 1 + 1 = I + I = (put those together side by side) --> II 3) = II 4) Now convert back to arabic numerals ----> 2, because II (Roman_) = '2' (Arabic) 5) So... in conclusion, 1 + 1 = I + I = II = 2. :P
In math you could actually do that, if you had an ismorphism F from the set of roman numbers to the set of natural numbers. Then: 1+1=F(I)+F(I)=F(I+I)=F(II)=2 The equation F(I)+F(I)=F(I+I) is true by the propertys of an Isomorphism.
I feel like it should be pointed out that the foundation of math is the fundamental logic of quantities. It's something even extraterrestrial life should be able to grasp, because no matter what symbols or words you use, the basics of quantity simply cannot be different.
The fundamentals of maths are absolutely not logic based off of quantities. The numbers we have 1,2,3,4……… are just labels of a result of a successor function. Yes we can use numbers to quantify amounts of real world things, however maths is not based off of that. Maths is based off of the relationship of many different variables. Which is why it’s so good for describing nature as everything in nature have a relationship. Extraterrestrial life wouldn’t get our maths because as I said our maths is more abstract and philosophical then just real world quantities. To boil the fundamentals of maths down to that would just be wrong
@@marshian__mallow2624And a successor function... is counting quantities. As a matter of fact, you can derive all of modern math from counting. It'll just take you a long, long time.
#5 is stated completely wrong. It should say if (1) and (2) are satisfied, then also (3), whereas: (1) 0 has property P (2) Whenever a has property P, so does s(a) [this is a CONDITION, not a consequence!] (3) Everey natural number a has property P.
1 + 1 is addition. Addition is increasing number a by number b Example: a = 1 b = 3 a + b = 1 + (1 + 1 + 1) The number outside bracket is a The number of the numbers in the bracket is b Back to the question: a = 1 = b a + b = 1 + (1) So we're incresing a(1) by b(1) So the answer is 11
This seems to be more about the meanings of each mathematical element. "Do things actually mean what we say or believe they mean?"" and "Must they?" There are so many presumptions about what numerical values should mean. "One what?" or "One first what plus one second what?" What establishes the seperateness of these 2 ones if they are already congruent? Maybe it should be established that numerical values in mathematical problems must/should be congruent. Also the act of addition, or operation, implies a time factor involved. 1+1=2, but what if the second one is added so much later, that they no longer correspond it the completion of the operation? The moments that the first one and the second one are presented to the problem may have been congruent in each of their moments, but at the time the second one is added, the first may have changed so much that they are no longer compatible. In which case we would have '1+1=1+1, and not 2'. So besides being congruent, numerical values in basic math operations are usually presumed to be descrete and timely (homogenous objects not in flux and dealt with instantly.) Another way to approach this is "If a person were to be presented with an addition problem for the very first time where they may be expected to understand it by applying it to some real world situation, then how might they end up misinterpreting it?
Nice video, I'm glad there are people to worry about the foundations : ) I'm not sure that's a good idea to not tell you chose an arbitrary set of axioms (Peano), and since you didn't choose Set Theory maybe it's a bit weird to talk about sets, afaik sets are not defined by Peano axioms. In your proof, you didn't explain how you manipulated the equal symbol, nor made a difference between the definitional and the propositional equalities. I understand the video wasn't meant to be that precise but then I don't get the point to explain the equality symbol before if not to use it afterwards. Finally, I spotted a few errors in the Peano axioms (#3 and induction) but I saw someone already told you in the comments. Anyway as I said, very nice video! Thank you! : )
This is a common confusion. Peano Axioms and PA (Peano Arithmetic) are two distinct things. Peano Axioms do indeed talk about sets of naturals, meanwhile PA is a first-order theory that doesn't mention sets of naturals at all, and only deal with natural numbers themselves. The distinction is that you cannot put Peano Axioms on a computer, while you can the first-order theory of Peano Arithmetic
Nice Video. A few things I want to point out: Natural Numbers #2 -> It is important to point out that every number has exactly 1 successor. #3 -> What you are saying is: 0 is not a successor but what you are writing is: 0 has no successor. #5 -> I would word it a little differently (as per your definition 0 is no divisor of 9 and 2 is no divisor of 9 so 3 is no divisor of 9). If a has a property and s(a) also has that property, because of its succession then every s*(a) has that property
4:34 "There exists no number who's succeeding is zero" - Could have been worded better, but OK. 5:38 "But zero cannot have a successor" - Now that's just not true, is it? You should have said: 'Zero cannot _be_ a successor'.
Like, "There exists no number whose succeeding (number) is zero" (He could've said smth like "There exists no natural number whose successor is zero" As for "But zero cannot have a successor" Well .. there's no arguing about that it's not true at all 😅😂😂
@@alextheconfuddled8983 holy sh*t this is so true like when you have to figure out the lines of working to get the marks but you cant figure out what to write, so you just sit there questioning the applications of such working out and why you even need to know them, falling into a deeper despair with every passing second......
I would have suggested real numbers axioms for that instead of using an oversimplification of Peano Axioms. Interesting video, anyways. I’m not really fine with seeing a zero in N, I’m used to see it in N_0 but if I’m missing something I’m very curious to know
Beside Peano's 5th axiom which was already pointed out, there are a couple other correction to make: 1) In the beginning, you aren't defining what equality means, you're just stating some of his properties. The problem is that there are other equivalence relations which also satisfy those properties. The true definition of equality for natural numbers is actually Peano's 4th axiom, extended by those equivalence properties. 2) In the end, you say that by changing the numerical values in the proof, one can show that 2+2=1. I guess you simply chose a bad wording fo this, because no numerical value is changed anywhere. What's actually changing are the symbols that represent those numerical values. All in all, it was a nice try. You surely understand what's going on much better than the average student.
@@VinyJones2 the properties of equality are typically laid down one level 'above' those of axioms like Peano Arithmetic; equality is a *logically* symbol, and it's rules are defined as part of the logic, rather than a specific theory. One of these logical axioms is about substitutions, which states: if one has deduced both P[x/t] and t = s, one can deduce P[x/s]. P here stands for an arbitrary logical statement with a free variable x, and P[x/t] stands for 'replace occurrences of x with the term t'.
I would add in another step before showing 2+2=1. That would be to replace 1 and 2 with arbitrary shapes. Replace 1 with a circle, and 2 with a line segment, for example. Then you are making a point that these are just symbols. You can then replace the circle with “2”, and the line segment with “1”. Just symbols. But then to use math in the real world, you would have to attach a meaning to those symbols. So “oneness”, can be represented by the symbol 1, or a circle or “2” as long as the users all agree on what the symbols means. So the number of legs on a normal horse can be represented by 4, or IV, or 😀 as long as the users understand the meaning of the symbol. The rules of natural numbers and addition and equality don’t change, just the symbols used to express them. PS: The posts around the Internet where they add and multiply bananas and shoes, etc are meaningless because the definitions are not given and everyone can make up there own rules to get any answer you want.
@@usuraiopeppino you only need the 2nd axiom schema of equality. Edit: And the 1st axiom of equality and the 3rd axiom schema as well, but only to prove the symmetry property. For all x, y: x=y & phi(y|v) implies phi(x|v). (axiom schema on phi. "x|v" means "substitute x with v.") Logical and is commutative, so we rewrite OP's statement to (b=d and a + b = c) implies a + d = c. Assume equality is symmetric (that can be proven), so we now have (d=b and a + b = c) implies a + d = c. Let phi be "a + v = c." phi(b|v) is "a + b = c", and phi(d|v) is "a + d = c." Thus, for all d, b: d=b & phi(b|v) implies phi(d|v). So for all d, b: (d=b & a+b=c) implies a+d=c. Walk this statement backwards (symmetry property of equality and commutativity of logical and), and apply universal quantification, and we get for all a, b, c, d: if a+b=c and b=d, then a+d=c. Q.E.D.
This is like when your in fifth grade when you do weird “pre-algebra” math but you can just add it together and get the same number instead of doing all of that work
Realistically 1 + 1 = 2 can be described as (an odd number + an odd number = an even number) the definition of an even number is 2x for some x in a set of integers. the definition of an odd number is 2x + 1 for some x in a set of integers. so (an odd + an odd) would be (2x + 1) + (2x + 1) = 4x + 2 by taking out a 2 using algebra it becomes 2(2x + 1) It's important to note that it's already established that x is an integer. Regardless what you do to an integer like add or multiply by another integer, it stays as an integer. Thus because 2x + 1 is an integer, and 2(2x + 1) looks similar to 2x (which again is defined as an even number), it can be said that an odd number + an odd number = an even number. Therefore, 1 + 1 = 2.
At 4:22 there's a mistake. Rule #3 says that rule #2 is false for 0. But 0 DOES have a succeeding number (which is 1). You could rephrase rule #2 like this: every number IS (instead of has) a succeeding number OR every number has a PRECEDING (instead of succeeding) number. Then rule #3 would be correct because 0 has no preceding number and is no successor of any number in N.
@@williamhorn411 oh, now that you mention it I remember I actually realised that while watching the vid xD, guess I couldn't recall it out of the context of the vid xD
It's not just math, but philosophy. 1 is not just a number, but a transcendental concept of unity that can be manipulated in certain ways, such as creating 2
Well, to be fair to Whitehead and Russell, before proving 1+1=2, they had to rebuild the foundations of math, since Russell-with his eponymous paradox-had just blasted Cantor's 'naïve' set theory to smithereens. As Carl Sagan said, *_“If you wish to make an apple pie from scratch, you must first invent the universe.”_* And that's what W&R did in their 3-volume _Principia_ (1910-1913). Ironically, proving 1+1=2 is fairly trivial using the axioms of _Peano Arithmetic_ (ca. 1889), as this video shows. So when W&R proved the same thing within their reformulated set theory, what they were really doing is showing that it is consistent with arithmetic; that there are no nasty surprises of the kind that destroyed Cantor's set theory. Today, most of us use ZFC (Peano axioms can be derived from ZFC), but it's not the only one. Besides, There's growing interest in *Category Theory* as well. There's no one true foundation for math-there are several. However, the critical takeaway is that they all should give results that are consistent with our everyday experience. Your axioms _must_ be able to prove that 1+1=2, otherwise they won't be of any practical use-and your grant application will most likely get rejected as well. 😜 That should also address your complaint that standard K-12 education doesn't teach us the foundational axioms of math. Furthermore, it's unnecessary. Seriously. School children are already overburdened with theory, and yet, are facing life unable to calculate change or understand the devastating impact of compound interest on credit card debt.
An argument is made that Cantor's "naive" set theory was not destroyed, or blasted apart by anthing, and certainly not by Russell. Russell's paradox never affected Cantor's theory because Cantor did not in fact have a principle of unlimited comprehension, as Frege did. (Frege was the target of Russell's paradox. It was Frege whose system was destroyed.) More generally, the logicists Russell and Frege took predicates to be primary and thought of sets as defined as the extension of a predicate. Only then did Russell realise that some apparently well-defined predicates (like "not being a member of itself") could not define a consistent set. Cantor, to the contrary, took sets (and particularly well-ordered sets) as primary. And he knew from early on that there were "inconsistent multiplicities" which could not be considered sets. Realising this avoids paradoxes. (My main source for this is Understanding the Infinite by Shaughan Lavine, but other sources also debunk the old idea that Cantor's theory was too naive to work, and needed to be fixed by logicians.)
Just use a number line. It is all about jumping !! By definition numbers are just an infinite series on a number line (we just define it to be). The addition function is simply a method used to jump to different numbers in that number line. With 1 + 1 the first number is where you start from. The number after the + signifies how many jumps to make, so here we have 1 jump and we arrive at 2. Or say 2 + 3 would start at 2 and make 3 jumps and arrive at 5. We are not really proving anything, rather we have decided to DEFINE a set of numbers in a prescribed order and the + operation is a jump to get you from a starting point to an end point. Similarly a - (subtraction) is backwards jumping. A multiply is starting at 0 and saying how many jumps and how big a jump. So 2 x 3 is saying start at 0 and we will do 2 jumps each of size 3. First jump gets us to the number 3 and second jump arrives at the destination of 6. (Or equally we could say 3 jumps each of size 2). A divide starts at the number and jumps to 0. E.g. 20 ÷ 5 says start at the number 20 on our line and make backwards jumps of size 5 and count how many jumps you made to get to 0. So you go to 15, 10, 5 and 0 making 4 jumps. If it was harder say 21 ÷ 5 you do the same and get to 1 after 4 jumps (16, 11, 6, 1) so to get to 0 you need a further mini jump of size 1/5, so in total you did 4 1/5 jumps (or 4.2) to get to 0.
It is based upon some definitions. In the proof how can be say that a+S(a)=S(a+b) If its just a definition of addition then why can't we just say 1+1=2 and move on.
I was once a senior physics student who thought Analysis would be helpful. Never did worse in a class. Never understood less. Never had a worse teacher. Lol. Got a gentleman's C.
The number 5 definition of the natural number would consider that if 0 is even, and a is even too, S(a) should be even but we know it is odd, does that mean that not any property shared by 0 and a propagates or that parity is not considered a property, and thus, what can be considered as property
I was a math major first in college (like 20 years ago) and one of my first classes at a 4 year university was proofs. I think we started with proving 1+1=2. This was the day I lost much joy(and some respect) for math and soon after changed my major to computer science. I wish they thought it more like this. I would have even started out that this does not really need to be proved (even though it does), but we are going to do use this as a simple teaching example...because when someone loves math and they cannot do a 1+1=2 example it makes them not feel that great.
I was first year physics (which I failed), and when the teacher started talking about this axiom insanity I raised my hand and said that 1 + 1 = 2 (or whatever was used) was merely putting beans together and describing the result. She asked back _"how does this work for 2√3"_ and that shut me up for long enough for them to ignore the issue. Ah computer programming ... nice indeed.
@@josboersema1352 - thanks for sharing. I did end up working in IT so somewhat CS related. I still like math topics though and watch it a lot of related videos. I'm still doing little math projects like a couple years ago I made a large number Sqrt function(for large numbers) called NewtonPlus. (I think it is the worlds fasted for C# and Java)
It seems to me that what happened here is that the definition of natural numbers is actually derived from the fact that 1+1=2, or at least that both of these (1+1=2 and the definition of natural numbers) don't have a logical direction between them; rather they are saying the same thing in different words. I'm not convinced by this that number sequences are more fundamental than addition.
Damn, I’m not into maths much but my dad always used to talk about how he had to prove this at some point in his mathematics degree, and I always wondered what he was talking about. Surprisingly, even if I’d just watched the Takeaway section at the end, I’d have understood how I was taking the Axioms for granted - and thus what he was on about. Thanks a lot for this! I can pull a rug over him next Christmas dinner (even if only for a moment before he pops me with a curve ball to make me the student once more). Ta!
6:51 The wording on the board isn’t quite right. It says “If 0 has some property that a also has then s(a) also has that property”. But as worded that’s not true. For example, if the property were “the number is a multiple of 2”, and a is 4, then both 0 and a are multiples of 2 but s(a) =5 is not. What the board should say is “If 0 has a property, and if a having a property implies s(a) also has the property, then all Natural Numbers have the property.” Which would be the induction that’s being talked about.
I remember these famous quotes. Science: explain something you don't understand. Philosophy:Confuse some basic fact in non understanding way. When we were 5, 6,7 we struggled to learn basic arithmetic operations. After years of struggling in arithmetic, algebra, geometry, we have to question the obvious arithmetic. I Believe the course is called pre Calculus
6:14 This axiom is formulated wrong. What you were meaning to talk about only applies to a property shared by 0 and all other natural numbers, but since a is defined as any one natural number (not all natural numbers) your formulation implies that any property shared by 0 and another natural number (a) must also be shared by all other natural numbers greater than a. This is very obviously not the case: A number being even (or odd) is a property of that number. The property of “even” is defined such that a number is even if and only if the quotient of that number with a divisor of 2 is an integer. 0 has the property of being even because 0/2=0 which is an integer. Now let’s say say a=4. 4 also has the property of being even because 4/2=2 which is an integer. We happen to know that s(4)=5, and since 5/2=2.5, not an integer, therefore 5 is not even. Yet the axiom as you worded it suggests that for any property shared by 0 and any a, s(a) also has that property. Therefore, your axiom is false.
You don’t have to ~re-define equality to define 1+1=2. Any operation “+” ie. A + B -> C, is a function, is a homomorphism, is of the form: f( xRy ) = f(x) # f(y) - is an equality relation already. Equality has those properties (being a “strict partial order”). 1+1=2 still has to be defined outside of those properties *which is why you had “S”… It’s better to look at each number { “1”, “2”, “3”, … } as members belonging to a Family (specifically a Toset), or a Group could work too I imagine- or a Sequence- or Tree/ Fan / Heap of degree 1 - or Comb of width 1… does matter… Regardless of the defined structure, the linearity of that structure, just apply that base operation of incrementation, and numbers will naturally appear. Families (usually sequences) are sometimes thought to require the indexing-by natural numbers idea,, so,, some of these structures lead to paradoxes. You could even consider a number a subclass of Tensor / Scalar, which naturally inherits Tensor Algebra (that defines its definition). So, proving 1+1=2 is kinda a waste of time, because the axioms are inherited surely, in whichever structure Numbers belong to, as members of that Structure. Cheers
I strongly disagree. Mathematics is not based on axiomatics, as postmodern formalists claim. The ontology of mathematics is non-linguistic or pre-linguistic, accessible through intuition. The most foundational mathematical relation we can express in language is the more-less relation, or rather the relation of increasing-decreasing, which we can formally write as the relational operators < and >. Equality.can be derived and defined from relational operators as the negation 'neither increases nor decreases', in other words: If A is neither more nor less than B, then A = B; in a suitable context of comparison.
@4:30, #3 incorrectly states what is meant. If #2 is false for zero then zero has no succeeding number, so there is no number one and the whole thing collapses in a heap. In this simplification #3 is redundant because of #1. Axioms are a program for a 1-pass human compiler.
There is a version of the trolley problem where a runaway trolley is going down a track with no one on it. But if you pull a lever you can deftly redirect the train to another track where there is 1 person, followed by 2 persons, followed by 3 persons and so on. So do you pull the lever to save 1/12 life?
6:30 i believe you oversimplified this a bit. The 5th peano axiom says that if 0 has that property and if from a having this property it follows that S(a) also has that property, then every natural number has that property.
Idk if this can answered here but its related to ref., sym., and trans properties. So for a relation on the real numbers, xRy iff (x/y) is rational. It's this an equivalence relation?
Equality isn't defined as such. Actually, if you define eqality only as a relation, without subtitution, there is a ton of thing you cannot proove. There is 2 acioms for equality. Reflexivity, and Libnotz substitution : if a=b, then for all prppsition, P(a) is equivalent to P(b) This is very important. Defining equality as a simple relation doesn't allow you to "replace" element.for example, you cannot prove that if x= f(x), then x=f(f(x)). Without substitution, there is no direct link beetween "f(f(x))" and "f(x)".
The thing you say about math at the end might not be true - or, put otherwise, there's probably a way of describing numbers using a more rigid (bitwise) structure which is less dependent on axioms.
This is part of formally defining a system that follows our base intuitions about how the physical world works, not proving anything, which then leads you to realize it's not that weird that math works well at describing the universe. Math is based on how it works after all.
![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
I've noticed a few word choices or errors that makes the statements confusing or incorrect so I'll put them here:
4:26 When I say #2 is false for 0, it would mean 0 has no successor (It does: it's 1). Rather, I should have phrased it that no number's successor is 0. I hope this is clear with the explanations thereafter.
5:38 -successor- predecessor
6:19 It should have said that if 0 has some property, and if a having with that property implies S(a) also has that property, all natural numbers have that property.
Sorry for these errors I hope you don't get confused by it.
If you are here as part of the SoME1 peer review, thank you so much for taking your time to watch this video.
If anyone is really low on time, you can skip to 8:58 or 11:11 where I prove, and give a conclusion on the video!
Thank you so much for watching!
I believe your correction of the fifth axiom is still inexact. You didn't quantify "a" in the first place, so it's really not clear what it means. It could either mean:
- for every a in N, if 0, a, s(a) share a property, then all natural numbers have that same property.
That's not the fifth axiom however. Moreover, it's false in natural numbers. For instance: 0, 1, s(1), all have the property of being less than 3. Clearly not all natural numbers are less than 3.
or it could mean:
- if 0 has a property, and for every a in N, a and s(a) have that property, then all natural numbers have that property. At this point it just becomes equivalent to "if, for every a in N, a has a property, then all natural numbers have that same property", which clearly is of little use.
The correct phrasing should be something like:
if 0 has a property, and if it's true that the same property belongs to s(a) if it belongs to some a, then all natural numbers share that property
@@betterpkm if P(0) and [P(a) => P(s(a))] then P(n) for all n
@@betterpkm Ah. That makes sense, it's induction. Okay, I had interpreted it as your first one, and was seeing a lot of properties that do not hold.
pin your comment too
@@betterpkm i think the clearest restatement would be to change the order of everything:
'given some property such that, some a having it requires that its s(a) has it, then if 0 has that property, therefore it is one which all natural numbers share.'
(or, less concisely:
'given some property such that, some natural number a having it requires that its immediate successor i.e. s(a) has it, which, consequently all further successors will have it, then if 0 i.e. the initial natural number has that property, therefore it is a property which all natural numbers share since all natural numbers besides 0 ultimately succeed from 0.')
When you go back to level 1 to do it in expert mode.
Smurfing
@@manetho5134 la la la la la la
Wtf kris dremur 😳
@@Just_me_lol yes?
@@HitTheFloor16 wait why do you ave dreamur en your name?your are part of the family but you are not a goat?
What apens ther
Thanks man this finally helped me get out of first grade; I've been stuck there for like 10 years now lol
Pro tip: learn calculus because you get to high school, it’s fun
GOLDEN HUMOR
Are u a zoomer
wait how much is 10 years again? 1?
@@Tise2033 yeah
The lecture: Proof for 1+1=2
The homework: Prove that 7+13=20
The brain teaser in the book: Prove that 3-2=1
The test: Prove that 3*4=12
*The exam: Prove the Collatz conjecture using only successors, 0 and 1.*
Prove that 9+10=19
@@minibelt3222 9+10=21* you mean
The life: prove that 1=2
@@minibelt3222 its 21 smh
😂😂....this is the most under-rated comment I ever saw...
Thanks mate my child had some trouble keeping up in their maths classes, this video has helped them a lot. Could you make another video proving that 1+2 = 3 soon?
he already proved 2+2=1, what more do you want?! :D
It seems like there's potential for a long series of videos!
We have, S(0) + S(S(0))
=> S(S(0) + S(0))
=> S(S(0+S(0))
=> S(S(S(0)))
=> 3
123 likes
I have a great proof for this fact, unfortunately, it will not fit in the TH-cam comments section.
I like the use of the equal sign to explain how an equal sign works
This whole proof reminds me of freshman year in school with some mind-altering substances, and suddenly wondering "What if C-A-T really spelled dog???"
"1+1 = 2 because 1 = 0+1 and 0+1+1 = 2. But if we define the successor to 0 as 一 then we'd say 0 + 一 = 一 and 0 + 一 + 一= 二 !!"
@@mallninja9805bro I don't know if you know chinese, but 一 is one and 二 is two. I don't know why I laughed so hard
Well said !
@@antequeragames5554Same in Japanese.
However, in both languages the old symbol for Four (亖) was replaced and it's all down hill from there 🥳
@@SolDizZo Oh shit are we having a party in here??
"1+1=2"
"But how do we prove that?"
"Thats the neat part , You don't"
Man your name gets me so confused
@@samisaac3908 damn you made me look now im in an existential crisis
@@CMDRunematti lol
What do you expect? The entire mathematics is made up fantasy. Forget about what physicists who don't know mathematics tells you.
@@aniksamiurrahman6365 mathematics is a way of looking at the world around you
This reminds me of the quote from Carl Sagan…
“If you wish to make an apple pie from scratch, you must first invent the universe”
Reminds me of assembly programming
@@Rudxain this guy codes
@@Rudxain reminds me of cpu designing...
@@Rudxain If you wish to play a game in Windows, you must first write the C compiler.
@@chaotickreg7024 real programmers write COBOL using a magnetic pen directly on the hard disk
POV: *It's your first day of Calculus class when the Math professor ask you to answer 1+1 plus with elaborate and concise explanation*
Elaborate and concise are antonyms.
@@Eagle0600 Not necessarily
This is, ironically, a bit higher level than calculus lol
I would say,. Add that 1 to another 1 so. 1 1 thats two 1 Therefore, 1+1 =2
1 2
@@Eagle0600 true you can be elaborately concise
i always love how complicated proofs for basic arithmetic is.
Thanks for this video! I was so confused when my kindergarten teacher said 1+1=2. I had no idea what it meant, but now I finally understand.
I always thought this was one of those topics that required either 2 seconds or +10 hours of discuss with no inbetweens, this video is almost a miracle.
The proof that 1+1=2 in Principia Mathematica is 360 pages long
@@XkinhoPT But can they prove that it's 360 pages?
@@Johannesburgus Now you're asking the big questions
Me who learned it in 5 hours back in preschool: Ha get fricked
@@Johannesburgus just count them duh
great explanation! im about to finishe my undergrad in mathematics and i've never really dove this far into things. good stuff, keep it up!
Thank you!! All this support gives me so much motivation to keep going!!!
thats sad actually
@@igorbispo8206 if only there was a word to describe this kind of moment
You're finishing undergrad and never learned addition??
@RatioKing kids these days
As a physics major, I sometimes think mathematicians overthink things.
nah, you're overthinking it
Typical physics student
Strange if you are scientist
Math is more enjoyable than Physics tho, even tho I never used most of the things that I learned on my life (or to be exact I don't even know what they are used for)
@@gmdascensia I beg to differ...
I always wondered how you could prove that 1 + 1 = 2 because that in of itself always seemed so fundamental to me. I mean, like you said, it's the first thing ever taught when kids are first learning math. I could not comprehend how the hell you could get any more basic until I watched this video. Thank you
this first symbol is called "one" and represents loneliness
the second symbol is called "plus" and represents the end of loneliness
the third symbol is called "one" and represents intrusion,
the fourth symbol is called "equal" and represents change ,
the fifth symbol is called "two" and represents the mistake
since 1+1=5
Ah yes, the Confucian proof
@@mallninja9805average day in Taiwan
1+1=5 only if you don't use a condom.
Now prove that 3 blues is approximately 1 brown
I see you are a man of culture
Well any two numbers an be approximately equal to eachother.
1~100~100000~pi
Same things holds true for arbitrary objects
3 blues ~ 1 brown
@@pedrosso0 wth does that mean
@@wosandakeweenjayaweera10-h48 if lets say were dealing with a massive number like 10^100, or larger, all numbers can essentially seem like 0 to that number, and hence you can say they are approximately each other in reference to said large number
@@wosandakeweenjayaweera10-h48 its the equivalent of saying "you know what, 1+1=3, which is approximately 2"
9:31 the part _"and using the fact that any a + 0 = a, we can cancel this 0"_ blew my mind
Ikr I am confused
Well 1+0=1?
@@snekky3415 putting like that makes me look like a kindergartener
@@snekky3415 9:31 was the moment i undertood why he made those assumption
@@o_enamuelThere are always cool basic things that make me rethink, like 5-2=3, but counting from 2,3,4,5 gives 4 numbers. (This was an important realization for off-by-one errors in programming)
And 100^2 is 10000, but any number less than 1, squared, becomes exponentially smaller, like 0.5^2=0.25
When you finished a game and go to the starting area with all your op gear
and its for some reason harder than everything you did before
eh, more like going to the hardest difficulty after finishing
You can use the 3 axioms of Peano, where the inductive axiom (technically axiom scheme) has "properties" instead of "subsets", and a hidden 0th axiom that governs the use of "=" and basically tells us that "successor" is a function:
0=0
Axiom scheme: m=n => S(m)=S(n), where m,n are natural number variables
Together with the 3rd axiom, this proves that any symbol for natural numbers is equal to itself; and I think that, more generally, all three properties of "=" are now theorems, using again the 3rd axiom. Maybe. Yes, I tried doing it and I think I need axioms for "replacing formulas" now, like in λ-calculus.
1 plus 1 is 10 because I used the binary system instead of the decimal system that we normally use. 10 has the first digit marked as 1 so we need to add 2 because the binary system multiplies it self by 2 every digit from right to left. 1001001 is 73 because the last digit is marked as 1 indicating that it is an odd number. 1 plus 8 plus 64 is 73. Search this topic for more information.
i mean, that doesn't change anything since 2 and 10 are two ways to represent the same quantity using different symbols. we can say that 3,1415926... = π because "π" is a way to represent the value and not the value itself.
@@pdd5793 True.
thought this was a shitpost.
It was not a shitpost.
Same! First I wanted to watch it just for lolz, but it was actually really interesting to me. I don't regret it! :D
It says a lot about natural languages that self-contradicting statements are not paradoxical.
For the longest time I believed that 1+1=11. But now I’m convinced that I’m wrong, and it is 2. Time to say sorry to my 2 year old cousin that I taught him the wrong equations
When you learn JavaScript instead of mathematics
@@epuntus😂😂
11 is just how you write 2 in base 1.
@@EdKolisbase 1? ah yes tally marks without the useful part of tally marks
I think you got the Peano axioms for natural numbers wrong. You say that:
1. 0 exists
2. For every natural number a, the successor of a, S(a), exists
3. Axiom 2 is false for 0.
But that's a contradiction! I think you meant that:
3. 0 is the successor of no natural number
Also, I'm not sure that axiom 5 defines the natural numbers. You could have the property "a is even" (whatever that means at that point), which contradicts your axiom, because S(a) would be odd for every natural number.
The Peano axiom is:
5. If 0 is in a set A, and if for every natural number a in A, S(a) is in A, then A = N
I think you tried to simplify this axiom, but I don't think it can :/
That's true. Thanks for letting me know!
Isn't your 5 an embedding of N in A? By your definition you could have elements in A that are not natural numbers and you would still incorrectly conclude that A = N, when N is a subset of A.
@@tricky778 Indeed, technically it is. However, this embedding is simply a bijection (actually the identity) from N to A by the axiom. It defines N as the only set of numbers that follow axioms 1-4.
It is not a conclusion, it is a definition
Assume there was a number a
in A that could not be reached by successive applications of S from 0. Then you could construct a strictly decreasing sequence starting from a that never reaches 0, which we know is impossible in N.
@@amarasa2567 oh, that wasn't an alternative to the collection of the other axioms but an additional one? Or 5 is missing the statement that 'forall a in A st. a =\= 0, exists b in A st. a = S b'?
@@tricky778 that looks like an equivalent formulation, though it arises from slightly less elementary notions (you basically used the basically the predecessor)
This feels like going back to play the Tutorial after finishing the Game
In New Game+, and there's a secret portal in the Tutorial Area to fight a Superboss called "Foundations".
@@nektariosorfanoudakis2270 lol
So it is an agreement that addition is a movement in a defined sequence of some symbols (numbers). Part of the agreement is that "+1" operator returns directly the next number in the sequence. So if we have "standardized" sequence of natural numbers: 1, 2, 3, 4, 5, ... that means 1+1=2. So in a nutshell it is an agreement about meaning of those symbols ( = and + included).
Russell and Whitehead took 426 pages, (I think) to reach 1+1=2 in Principia. That excerpt ain't gonna tell you squat.
And there was a mistake in it, right?
@@synchronium24 there was no mistake but it was shown when they were almost done writing that there can never be axioms that describe all of math
@@lukedavis6711 Thanks, guess I was wrong. I take it you are referring to Godel's Incompleteness Theorem?
[Disclaimer: I'm quoting myself from a longer comment I made elsewhere. Not that I love repeating myself, but rather, your comment seemed to imply that Whitehead and Russel were unnecessarily verbose in their proof. So I thought I'd set that straight.]
Well, to be fair to Whitehead and Russell, before proving 1+1=2, they had to rebuild the foundations of math, since Russell-with his eponymous paradox-had just blasted Cantor's 'naïve' set theory to smithereens.
As Carl Sagan said, *_“If you wish to make an apple pie from scratch, you must first invent the universe.”_* And that's what W&R did in their 3-volume _Principia_ (1910-1913).
Ironically, proving 1+1=2 is fairly trivial using the axioms of _Peano Arithmetic_ (ca. 1889), as this video shows. So when W&R proved the same thing within their reformulated set theory, what they were really doing is showing that it is consistent with arithmetic; that there are no nasty surprises of the kind that destroyed Cantor's set theory.
@@synchronium24 yea I'm not an expert myself but that's the story as I understand it. Russell was trying to make sure that self reference was impossible in this system to prevent what happened to naive set theory, but godel showed that you can force self reference on any system that's sufficiently powerful (arithmetic being one); the problem being that with self reference you can create the liars paradox.
It took me quite a while to convince myself this wasn’t just an elaborate meme
Ahahah
No this is legitimately an introductory course to a real field of math called first order logic.
Even easier:
1) Use roman numerals
2) 1 + 1 = I + I = (put those together side by side) --> II
3) = II
4) Now convert back to arabic numerals ----> 2, because II (Roman_) = '2' (Arabic)
5) So... in conclusion, 1 + 1 = I + I = II = 2.
:P
In math you could actually do that, if you had an ismorphism F from the set of roman numbers to the set of natural numbers.
Then:
1+1=F(I)+F(I)=F(I+I)=F(II)=2
The equation F(I)+F(I)=F(I+I) is true by the propertys of an Isomorphism.
You need to
Prove I+I=II
@@geraldvaughn8403 using the fundamental theorem of calculus, and using the limit of sums, we can find the limit of I+I as I approaches 1, which is II
٢؟
cool so 1 + 5 = 4 as well right?. (I + V = IV)
I feel like it should be pointed out that the foundation of math is the fundamental logic of quantities. It's something even extraterrestrial life should be able to grasp, because no matter what symbols or words you use, the basics of quantity simply cannot be different.
The fundamentals of maths are absolutely not logic based off of quantities. The numbers we have 1,2,3,4……… are just labels of a result of a successor function. Yes we can use numbers to quantify amounts of real world things, however maths is not based off of that. Maths is based off of the relationship of many different variables. Which is why it’s so good for describing nature as everything in nature have a relationship. Extraterrestrial life wouldn’t get our maths because as I said our maths is more abstract and philosophical then just real world quantities. To boil the fundamentals of maths down to that would just be wrong
what? we dont found math in arithmetic
@@marshian__mallow2624And a successor function... is counting quantities. As a matter of fact, you can derive all of modern math from counting. It'll just take you a long, long time.
Virgin mathematical explaination
Vs
Chad: If I have one apple and I get one more apple I will have two apples
Bruh
Omega Chad: 1+1=1 because one protein shake plus one protein shake is one big protein shake
@@azai.mp4 a big "1"
@@SirDrakeFrancis A 1 that's 2 times the size of the shakes you started with, even.
999IQ - If u apple an apple and u have a pen u get PEN PINEAPPLE APPLE PEN
I opened this expecting a shitpost based on how 3Blue1Brown presents their videos, but instead I got a really neat TEDTalk
#5 is stated completely wrong. It should say if (1) and (2) are satisfied, then also (3), whereas:
(1) 0 has property P
(2) Whenever a has property P, so does s(a) [this is a CONDITION, not a consequence!]
(3) Everey natural number a has property P.
1 + 1 is addition. Addition is increasing number a by number b
Example:
a = 1 b = 3
a + b = 1 + (1 + 1 + 1)
The number outside bracket is a
The number of the numbers in the bracket is b
Back to the question:
a = 1 = b
a + b = 1 + (1)
So we're incresing a(1) by b(1)
So the answer is 11
This seems to be more about the meanings of each mathematical element. "Do things actually mean what we say or believe they mean?"" and "Must they?"
There are so many presumptions about what numerical values should mean. "One what?" or "One first what plus one second what?" What establishes the seperateness of these 2 ones if they are already congruent? Maybe it should be established that numerical values in mathematical problems must/should be congruent. Also the act of addition, or operation, implies a time factor involved. 1+1=2, but what if the second one is added so much later, that they no longer correspond it the completion of the operation? The moments that the first one and the second one are presented to the problem may have been congruent in each of their moments, but at the time the second one is added, the first may have changed so much that they are no longer compatible. In which case we would have '1+1=1+1, and not 2'. So besides being congruent, numerical values in basic math operations are usually presumed to be descrete and timely (homogenous objects not in flux and dealt with instantly.)
Another way to approach this is "If a person were to be presented with an addition problem for the very first time where they may be expected to understand it by applying it to some real world situation, then how might they end up misinterpreting it?
Nice video, I'm glad there are people to worry about the foundations : )
I'm not sure that's a good idea to not tell you chose an arbitrary set of axioms (Peano), and since you didn't choose Set Theory maybe it's a bit weird to talk about sets, afaik sets are not defined by Peano axioms.
In your proof, you didn't explain how you manipulated the equal symbol, nor made a difference between the definitional and the propositional equalities. I understand the video wasn't meant to be that precise but then I don't get the point to explain the equality symbol before if not to use it afterwards.
Finally, I spotted a few errors in the Peano axioms (#3 and induction) but I saw someone already told you in the comments.
Anyway as I said, very nice video! Thank you! : )
This is a common confusion. Peano Axioms and PA (Peano Arithmetic) are two distinct things. Peano Axioms do indeed talk about sets of naturals, meanwhile PA is a first-order theory that doesn't mention sets of naturals at all, and only deal with natural numbers themselves. The distinction is that you cannot put Peano Axioms on a computer, while you can the first-order theory of Peano Arithmetic
@@gchtrivs7897 Oh alright, thank you a lot for the clarification!
0:38 I laughed so hard when I saw that you had drawn the 3b1b pi figure by hand. And I was fascinated when you were drawing it at 11:10.
What does it mean?
@@ararix3722 it's a mascot of the excellent mathematical TH-cam channel, 3Blue1Brown
This is what teacher want us to write when do the questions
"Remember to show all the work!"
ALL the work!
*ALL!!* the work!
Yeah and we have to write this in 2 lines
“But I wrote all my work down!”
“No, I meant ALL work done. I wanna know the history of this mathematical equation.”
I am your 10,000th subscriber. Congrats.
Nice Video. A few things I want to point out:
Natural Numbers
#2 -> It is important to point out that every number has exactly 1 successor.
#3 -> What you are saying is: 0 is not a successor but what you are writing is: 0 has no successor.
#5 -> I would word it a little differently (as per your definition 0 is no divisor of 9 and 2 is no divisor of 9 so 3 is no divisor of 9). If a has a property and s(a) also has that property, because of its succession then every s*(a) has that property
#2 ....."1"is confusing; properly "a single"is understandable.
I was expecting a silly 3blue1brown parody but it's actually just a serious explanation. Amazing
would have preferred that instead
4:34 "There exists no number who's succeeding is zero" - Could have been worded better, but OK.
5:38 "But zero cannot have a successor" - Now that's just not true, is it? You should have said: 'Zero cannot _be_ a successor'.
He wrote a comment where he fixed that second one, it just isn't pinned so you might have to do some digging
Like, "There exists no number whose succeeding (number) is zero"
(He could've said smth like "There exists no natural number whose successor is zero"
As for "But zero cannot have a successor"
Well .. there's no arguing about that it's not true at all 😅😂😂
Wow thank you so much, my teacher has been forcing me to right the proof so many times on the 1+1 question, finally I can pass my exam this year!!
"work out 1+1"
*"show your working (20 marks)"*
@@alextheconfuddled8983 holy sh*t this is so true like when you have to figure out the lines of working to get the marks but you cant figure out what to write, so you just sit there questioning the applications of such working out and why you even need to know them, falling into a deeper despair with every passing second......
3:24 “we need to define natural numbers. NUMBER ONE” 💀💀💀💀
I would have suggested real numbers axioms for that instead of using an oversimplification of Peano Axioms. Interesting video, anyways. I’m not really fine with seeing a zero in N, I’m used to see it in N_0 but if I’m missing something I’m very curious to know
Beside Peano's 5th axiom which was already pointed out, there are a couple other correction to make:
1) In the beginning, you aren't defining what equality means, you're just stating some of his properties. The problem is that there are other equivalence relations which also satisfy those properties. The true definition of equality for natural numbers is actually Peano's 4th axiom, extended by those equivalence properties.
2) In the end, you say that by changing the numerical values in the proof, one can show that 2+2=1. I guess you simply chose a bad wording fo this, because no numerical value is changed anywhere. What's actually changing are the symbols that represent those numerical values.
All in all, it was a nice try. You surely understand what's going on much better than the average student.
point 1) seem to be on the way of solve what bother me : how you prove that (a + b = c and b = d ) => a + d = c
@@VinyJones2 yes, induction on b should do the trick.
@@VinyJones2 the properties of equality are typically laid down one level 'above' those of axioms like Peano Arithmetic; equality is a *logically* symbol, and it's rules are defined as part of the logic, rather than a specific theory. One of these logical axioms is about substitutions, which states: if one has deduced both P[x/t] and t = s, one can deduce P[x/s]. P here stands for an arbitrary logical statement with a free variable x, and P[x/t] stands for 'replace occurrences of x with the term t'.
I would add in another step before showing 2+2=1. That would be to replace 1 and 2 with arbitrary shapes. Replace 1 with a circle, and 2 with a line segment, for example. Then you are making a point that these are just symbols. You can then replace the circle with “2”, and the line segment with “1”. Just symbols. But then to use math in the real world, you would have to attach a meaning to those symbols. So “oneness”, can be represented by the symbol 1, or a circle or “2” as long as the users all agree on what the symbols means. So the number of legs on a normal horse can be represented by 4, or IV, or 😀 as long as the users understand the meaning of the symbol. The rules of natural numbers and addition and equality don’t change, just the symbols used to express them.
PS: The posts around the Internet where they add and multiply bananas and shoes, etc are meaningless because the definitions are not given and everyone can make up there own rules to get any answer you want.
@@usuraiopeppino you only need the 2nd axiom schema of equality. Edit: And the 1st axiom of equality and the 3rd axiom schema as well, but only to prove the symmetry property.
For all x, y: x=y & phi(y|v) implies phi(x|v). (axiom schema on phi. "x|v" means "substitute x with v.")
Logical and is commutative, so we rewrite OP's statement to (b=d and a + b = c) implies a + d = c.
Assume equality is symmetric (that can be proven), so we now have (d=b and a + b = c) implies a + d = c.
Let phi be "a + v = c." phi(b|v) is "a + b = c", and phi(d|v) is "a + d = c."
Thus, for all d, b: d=b & phi(b|v) implies phi(d|v). So for all d, b: (d=b & a+b=c) implies a+d=c. Walk this statement backwards (symmetry property of equality and commutativity of logical and), and apply universal quantification, and we get for all a, b, c, d: if a+b=c and b=d, then a+d=c. Q.E.D.
This is like when your in fifth grade when you do weird “pre-algebra” math but you can just add it together and get the same number instead of doing all of that work
I love how there is a lot of actually helpul videos in this channel about math, but this is the most viewed.
that's why I don't upload my clips.
Realistically 1 + 1 = 2 can be described as (an odd number + an odd number = an even number)
the definition of an even number is 2x for some x in a set of integers.
the definition of an odd number is 2x + 1 for some x in a set of integers.
so (an odd + an odd) would be (2x + 1) + (2x + 1) = 4x + 2
by taking out a 2 using algebra it becomes 2(2x + 1)
It's important to note that it's already established that x is an integer. Regardless what you do to an integer like add or multiply by another integer, it stays as an integer.
Thus because 2x + 1 is an integer, and 2(2x + 1) looks similar to 2x (which again is defined as an even number), it can be said that an odd number + an odd number = an even number.
Therefore, 1 + 1 = 2.
At 4:22 there's a mistake. Rule #3 says that rule #2 is false for 0. But 0 DOES have a succeeding number (which is 1).
You could rephrase rule #2 like this: every number IS (instead of has) a succeeding number OR every number has a PRECEDING (instead of succeeding) number. Then rule #3 would be correct because 0 has no preceding number and is no successor of any number in N.
Never mind, I just read the explanatory comment further down. Maybe pin that to the top of the comment section?
Thanks for this really well broken down explanation!
Glad you enjoyed it!
4:19 I paused and stared at that for literally 30 seconds wondering “what does e+c mean???”
well, 'c' is usually just some constant
@@simdimdim It's not e+c, it's "etc", that was the joke lol
@@williamhorn411 oh, now that you mention it I remember I actually realised that while watching the vid xD, guess I couldn't recall it out of the context of the vid xD
"1+1"
Me: 2
Descartes: Are you sure?
Me: ...
Trust me, I've learned to question this sort of thing even since python once told me that "5" > "10" is true.
It's not just math, but philosophy. 1 is not just a number, but a transcendental concept of unity that can be manipulated in certain ways, such as creating 2
teacher: show your work
teachers expectations:
Well, to be fair to Whitehead and Russell, before proving 1+1=2, they had to rebuild the foundations of math, since Russell-with his eponymous paradox-had just blasted Cantor's 'naïve' set theory to smithereens.
As Carl Sagan said, *_“If you wish to make an apple pie from scratch, you must first invent the universe.”_* And that's what W&R did in their 3-volume _Principia_ (1910-1913).
Ironically, proving 1+1=2 is fairly trivial using the axioms of _Peano Arithmetic_ (ca. 1889), as this video shows. So when W&R proved the same thing within their reformulated set theory, what they were really doing is showing that it is consistent with arithmetic; that there are no nasty surprises of the kind that destroyed Cantor's set theory.
Today, most of us use ZFC (Peano axioms can be derived from ZFC), but it's not the only one. Besides, There's growing interest in *Category Theory* as well.
There's no one true foundation for math-there are several.
However, the critical takeaway is that they all should give results that are consistent with our everyday experience. Your axioms _must_ be able to prove that 1+1=2, otherwise they won't be of any practical use-and your grant application will most likely get rejected as well. 😜
That should also address your complaint that standard K-12 education doesn't teach us the foundational axioms of math. Furthermore, it's unnecessary. Seriously. School children are already overburdened with theory, and yet, are facing life unable to calculate change or understand the devastating impact of compound interest on credit card debt.
An argument is made that Cantor's "naive" set theory was not destroyed, or blasted apart by anthing, and certainly not by Russell. Russell's paradox never affected Cantor's theory because Cantor did not in fact have a principle of unlimited comprehension, as Frege did. (Frege was the target of Russell's paradox. It was Frege whose system was destroyed.) More generally, the logicists Russell and Frege took predicates to be primary and thought of sets as defined as the extension of a predicate. Only then did Russell realise that some apparently well-defined predicates (like "not being a member of itself") could not define a consistent set. Cantor, to the contrary, took sets (and particularly well-ordered sets) as primary. And he knew from early on that there were "inconsistent multiplicities" which could not be considered sets. Realising this avoids paradoxes.
(My main source for this is Understanding the Infinite by Shaughan Lavine, but other sources also debunk the old idea that Cantor's theory was too naive to work, and needed to be fixed by logicians.)
This new manim edition sure looks good
I've always believed that proving something in math isn't for me, and this video finally proved it...
Just use a number line. It is all about jumping !!
By definition numbers are just an infinite series on a number line (we just define it to be). The addition function is simply a method used to jump to different numbers in that number line.
With 1 + 1 the first number is where you start from. The number after the + signifies how many jumps to make, so here we have 1 jump and we arrive at 2.
Or say 2 + 3 would start at 2 and make 3 jumps and arrive at 5.
We are not really proving anything, rather we have decided to DEFINE a set of numbers in a prescribed order and the + operation is a jump to get you from a starting point to an end point. Similarly a - (subtraction) is backwards jumping.
A multiply is starting at 0 and saying how many jumps and how big a jump. So 2 x 3 is saying start at 0 and we will do 2 jumps each of size 3. First jump gets us to the number 3 and second jump arrives at the destination of 6. (Or equally we could say 3 jumps each of size 2).
A divide starts at the number and jumps to 0. E.g. 20 ÷ 5 says start at the number 20 on our line and make backwards jumps of size 5 and count how many jumps you made to get to 0. So you go to 15, 10, 5 and 0 making 4 jumps.
If it was harder say 21 ÷ 5 you do the same and get to 1 after 4 jumps (16, 11, 6, 1) so to get to 0 you need a further mini jump of size 1/5, so in total you did 4 1/5 jumps (or 4.2) to get to 0.
That's what i was trying to explain to my teacher in first grade, he didn't want to understand me.
Imagine we were taught mathematics this way back in elementary school
Y’all would lose your mind:)
Great work mate, I really love this video!
Thank you so much! I’m really glad you enjoyed!!!
It is based upon some definitions.
In the proof how can be say that
a+S(a)=S(a+b)
If its just a definition of addition then why can't we just say 1+1=2 and move on.
I was once a senior physics student who thought Analysis would be helpful. Never did worse in a class. Never understood less. Never had a worse teacher. Lol. Got a gentleman's C.
The number 5 definition of the natural number would consider that if 0 is even, and a is even too, S(a) should be even but we know it is odd, does that mean that not any property shared by 0 and a propagates or that parity is not considered a property, and thus, what can be considered as property
I was a math major first in college (like 20 years ago) and one of my first classes at a 4 year university was proofs. I think we started with proving 1+1=2. This was the day I lost much joy(and some respect) for math and soon after changed my major to computer science. I wish they thought it more like this. I would have even started out that this does not really need to be proved (even though it does), but we are going to do use this as a simple teaching example...because when someone loves math and they cannot do a 1+1=2 example it makes them not feel that great.
I was first year physics (which I failed), and when the teacher started talking about this axiom insanity I raised my hand and said that 1 + 1 = 2 (or whatever was used) was merely putting beans together and describing the result. She asked back _"how does this work for 2√3"_ and that shut me up for long enough for them to ignore the issue. Ah computer programming ... nice indeed.
@@josboersema1352 - thanks for sharing. I did end up working in IT so somewhat CS related. I still like math topics though and watch it a lot of related videos. I'm still doing little math projects like a couple years ago I made a large number Sqrt function(for large numbers) called NewtonPlus. (I think it is the worlds fasted for C# and Java)
If only they taught us that in school. This is something that should be made known to young learners.
They will not understand it + it is too abstract + it is complicated to explain why the axioms are not made up
It seems to me that what happened here is that the definition of natural numbers is actually derived from the fact that 1+1=2, or at least that both of these (1+1=2 and the definition of natural numbers) don't have a logical direction between them; rather they are saying the same thing in different words. I'm not convinced by this that number sequences are more fundamental than addition.
1+1=2?
Proo: let 1 = a&b ---1
a=b
a/b = 1/1
By componendo-dividendo
a+b/2a-b = 1+1/2-1
a+b=2
From equation 1
1+1=2
Hence, proved
Really tougth that it was a parody of 3blue1brown but it was actually a nice and relaxing demonstrarion of pure logic and math. 10/10
This comes to show you can make anything complicated in math
Damn, I’m not into maths much but my dad always used to talk about how he had to prove this at some point in his mathematics degree, and I always wondered what he was talking about. Surprisingly, even if I’d just watched the Takeaway section at the end, I’d have understood how I was taking the Axioms for granted - and thus what he was on about.
Thanks a lot for this! I can pull a rug over him next Christmas dinner (even if only for a moment before he pops me with a curve ball to make me the student once more). Ta!
6:51 The wording on the board isn’t quite right. It says
“If 0 has some property that a also has then s(a) also has that property”.
But as worded that’s not true. For example, if the property were “the number is a multiple of 2”, and a is 4, then both 0 and a are multiples of 2 but s(a) =5 is not.
What the board should say is
“If 0 has a property, and if a having a property implies s(a) also has the property, then all Natural Numbers have the property.”
Which would be the induction that’s being talked about.
I remember these famous quotes.
Science: explain something you don't understand.
Philosophy:Confuse some basic fact in non understanding way.
When we were 5, 6,7 we struggled to learn basic arithmetic operations.
After years of struggling in arithmetic, algebra, geometry, we have to question the obvious arithmetic. I Believe the course is called pre Calculus
This is a very enlightening explanation. This is one of those concepts that is incredibly difficult to explain.
When you go back to play the level 1 but in very hard mode.
Yep.. Bro.. I was making a rocket and thought how is 1+1 =2.. Helped a lot ♥️
6:14 This axiom is formulated wrong. What you were meaning to talk about only applies to a property shared by 0 and all other natural numbers, but since a is defined as any one natural number (not all natural numbers) your formulation implies that any property shared by 0 and another natural number (a) must also be shared by all other natural numbers greater than a.
This is very obviously not the case:
A number being even (or odd) is a property of that number. The property of “even” is defined such that a number is even if and only if the quotient of that number with a divisor of 2 is an integer. 0 has the property of being even because 0/2=0 which is an integer. Now let’s say say a=4. 4 also has the property of being even because 4/2=2 which is an integer. We happen to know that s(4)=5, and since 5/2=2.5, not an integer, therefore 5 is not even. Yet the axiom as you worded it suggests that for any property shared by 0 and any a, s(a) also has that property. Therefore, your axiom is false.
Can anyone please explain to me what the fifth definition on the narural numbers is about?
I'm foriegn so I'm not that good at Eng...
0:08
mans lookin like the original star walker
This math is pissing him off
this feels like watching the creation of the universe, like god thinking about how everything should work
Now prove that sin²(pi/4)=1/2.😂
Whether it is a variable or number plus means link. Attach. Structural definitions. Because two things are same and you can only have a link to say 2.
I didnt get it in the #3 concept at 4:37. How does #2 equals false?
Why does every lecture on the principles of mathematics end up as a philosophy lesson?
Because philosophy is the principle foundation of everything.
You can't talk about thinking without talking about thinking.
You don’t have to ~re-define equality to define 1+1=2.
Any operation “+” ie. A + B -> C, is a function, is a homomorphism, is of the form:
f( xRy ) = f(x) # f(y) - is an equality relation already.
Equality has those properties (being a “strict partial order”). 1+1=2 still has to be defined outside of those properties *which is why you had “S”…
It’s better to look at each number { “1”, “2”, “3”, … } as members belonging to a Family (specifically a Toset), or a Group could work too I imagine- or a Sequence- or Tree/ Fan / Heap of degree 1 - or Comb of width 1… does matter…
Regardless of the defined structure, the linearity of that structure, just apply that base operation of incrementation, and numbers will naturally appear.
Families (usually sequences) are sometimes thought to require the indexing-by natural numbers idea,, so,, some of these structures lead to paradoxes.
You could even consider a number a subclass of Tensor / Scalar, which naturally inherits Tensor Algebra (that defines its definition).
So, proving 1+1=2 is kinda a waste of time, because the axioms are inherited surely, in whichever structure Numbers belong to, as members of that Structure.
Cheers
You can also prove the second property using von Neumann ordinals and set union operations defining the successor function as s(a) = aU{a}.
I strongly disagree. Mathematics is not based on axiomatics, as postmodern formalists claim. The ontology of mathematics is non-linguistic or pre-linguistic, accessible through intuition. The most foundational mathematical relation we can express in language is the more-less relation, or rather the relation of increasing-decreasing, which we can formally write as the relational operators < and >.
Equality.can be derived and defined from relational operators as the negation 'neither increases nor decreases', in other words: If A is neither more nor less than B, then A = B; in a suitable context of comparison.
When people take skepticism almost way too far.
@4:30, #3 incorrectly states what is meant. If #2 is false for zero then zero has no succeeding number, so there is no number one and the whole thing collapses in a heap. In this simplification #3 is redundant because of #1. Axioms are a program for a 1-pass human compiler.
9:14-9:19 a is the whole expression but for b we only take the particle 0. this looks completely arbitrary and needs further elaboration.
I think it is badly explained:
define both a and b are zero. Insert them into a + s(b) --> s(0 + s(0))
However, why this is correct eludes me.
Now who, how and why defined these properties of Natural numbers, plus sign and equal sign is the question
There is a version of the trolley problem where a runaway trolley is going down a track with no one on it. But if you pull a lever you can deftly redirect the train to another track where there is 1 person, followed by 2 persons, followed by 3 persons and so on.
So do you pull the lever to save 1/12 life?
6:30 i believe you oversimplified this a bit. The 5th peano axiom says that if 0 has that property and if from a having this property it follows that S(a) also has that property, then every natural number has that property.
Idk if this can answered here but its related to ref., sym., and trans properties. So for a relation on the real numbers, xRy iff (x/y) is rational. It's this an equivalence relation?
Equality isn't defined as such. Actually, if you define eqality only as a relation, without subtitution, there is a ton of thing you cannot proove.
There is 2 acioms for equality. Reflexivity, and Libnotz substitution : if a=b, then for all prppsition, P(a) is equivalent to P(b)
This is very important. Defining equality as a simple relation doesn't allow you to "replace" element.for example, you cannot prove that if x= f(x), then x=f(f(x)). Without substitution, there is no direct link beetween "f(f(x))" and "f(x)".
The thing you say about math at the end might not be true - or, put otherwise, there's probably a way of describing numbers using a more rigid (bitwise) structure which is less dependent on axioms.
This is part of formally defining a system that follows our base intuitions about how the physical world works, not proving anything, which then leads you to realize it's not that weird that math works well at describing the universe. Math is based on how it works after all.
=> 1+1= 2
=> 1= 2-1
=> 1= 1
Hence, Proved.
ALSO
Only 1+1 is Equal 2 because,
If we put
1+1=3
=> 1= 3-1
=> 1≠ 2
We can prove any equation now...
Personality test: do you tend to overthink?
Mathematicians: