How do I convince someone that $1+1=2$ may not necessarily be true?

Question

Me and my friend were arguing over this "fact" that we all know and hold dear. However, I do know that $1+1=2$ is an axiom. That is why I beg to differ. Neither of us have the required mathematical knowledge to convince each other.

And that is why, we decided to turn to Math Stackexchange for help.

What would be stack's opinion?

Answer 1

It seems that you and your friend lack the mathematical knowledge to handle this delicate point. What is a proof? What is an axiom? What are $1,+,2,=$?

Well, let me try and be concise about things.

• A proof is a short sequence of deductions from axioms and assumptions, where at every step we deduce information from our axioms, our assumptions and previously deduced sentences.

• An axiom is simply an assumption.

• $1,+,2,=$ are just letters and symbols. We usually associate $=$ with equality; that is two things are equal if and only if they are the same thing. As for $1,2,+$ we have a natural understanding of what they are but it is important to remember those are just letters which can be used elsewhere (and they are used elsewhere, often).

You want to prove to your friend that $1+1=2$, where those symbols are interpreted as they are naturally perceived. $1$ is the amount of hands attached to a healthy arm of a human being; $2$ is the number of arms attached to a healthy human being; and $+$ is the natural sense of addition.

From the above, what you want to show, mathematically, is that if you are a healthy human being then you have exactly two hands.

But in mathematics we don't talk about hands and arms. We talk about mathematical objects. We need a suitable framework, and we need axioms to define the properties of these objects. For the sake of the natural numbers which include $1,2,+$ and so on, we can use the Peano Axioms (PA). These axioms are commonly accepted as the definition of the natural numbers in mathematics, so it makes sense to choose them.

I don't want to give a full exposition of PA, so I will only use the part I need from the axioms, the one discussing addition. We have three primary symbols in the language: $0, S, +$. And our axioms are:

1. For every $x$ and for every $y$, $S(x)=S(y)$ if and only if $x=y$.
2. For every $x$ either $x=0$ or there is some $y$ such that $x=S(y)$.
3. There is no $x$ such that $S(x)=0$.
4. For every $x$ and for every $y$, $x+y=y+x$.
5. For every $x$, $x+0=x$.
6. For every $x$ and for every $y$, $x+S(y)=S(x+y)$.

This axioms tell us that $S(x)$ is to be thought as $x+1$ (the successor of $x$), and it tells us that addition is commutative and what relations it bears with the successor function.

Now we need to define what are $1$ and $2$. Well, $1$ is a shorthand for $S(0)$ and $2$ is a shorthand for $S(1)$, or $S(S(0))$.

Finally! We can write a proof that $1+1=2$:

1. $S(0)+S(0)=S(S(0)+0)$ (by axiom 6).
2. $S(0)+0 = S(0)$ (by axiom 5).
3. $S(S(0)+0) = S(S(0))$ (by the second deduction and axiom 1).
4. $S(0)+S(0) = S(S(0))$ (from the first and third deductions).

And that is what we wanted to prove.

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Note that the context is quite important. We are free to define the symbols to mean whatever it is we want them to mean. We can easily define a new context, and a new framework in which $1+1\neq 2$. Much like we can invent a whole new language in which Bye is a word for greeting people when you meet them, and Hi is a word for greeting people as they leave.

To see that $1+1\neq2$ in some context, simply define the following axioms:

1. $1\neq 2$
2. For every $x$ and for every $y$, $x+y=x$.

Now we can write a proof that $1+1\neq 2$:

1. $1+1=1$ (axiom 2 applied for $x=1$).
2. $1\neq 2$ (axiom 1).
3. $1+1\neq 2$ (from the first and second deductions).

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If you read this far, you might also be interested to read these:

1. How would one be able to prove mathematically that $1+1 = 2$?
2. What is the basis for a proof?
3. How is a system of axioms different from a system of beliefs?

Answer 2

Those interested in pushing this question back further than Asaf Karagila did (well past logic and into the morass of philosophy) may be interested in the following comments that were written in 1860 (full reference below). Also, although Asaf's treatment here avoids this, there are certain issues when defining addition of natural numbers in terms of the successor operation that are often overlooked. See my 22 November 2011 and 28 November 2011 posts in the Math Forum group math-teach.

$[\ldots]$ consider this case. There is a world in which, whenever two pairs of things are either placed in proximity or are contemplated together, a fifth thing is immediately created and brought within the contemplation of the mind engaged in putting two and two together. This is surely neither inconceivable, for we can readily conceive the result by thinking of common puzzle tricks, nor can it be said to be beyond the power of Omnipotence, yet in such a world surely two and two would make five. That is, the result to the mind of contemplating two two’s would be to count five. This shows that it is not inconceivable that two and two might make five; but, on the other hand, it is perfectly easy to see why in this world we are absolutely certain that two and two make four. There is probably not an instant of our lives in which we are not experiencing the fact. We see it whenever we count four books, four tables or chairs, four men in the street, or the four corners of a paving stone, and we feel more sure of it than of the rising of the sun to-morrow, because our experience upon the subject is so much wider and applies to such an infinitely greater number of cases.

The above passage comes from:

James Fitzjames Stephen (1829-1894), Review of Henry Longueville Mansel (1820-1871), Metaphysics; or, the Philosophy of Consciousness, Phenomenal and Real (1860), The Saturday Review 9 #244 (30 June 1860), pp. 840-842. [see page 842]

Stephen’s review of Mansel's book is reprinted on pp. 320-335 of Stephen's 1862 book Essays, where the quote above can be found on page 333.

(ADDED 2 YEARS LATER) Because my answer continues to receive sporadic interest and because I came across something this weekend related to it, I thought I would extend my answer by adding a couple of items.

The first new item, [A], is an excerpt from a 1945 paper by Charles Edward Whitmore. I came across Whitmore's paper several years ago when I was looking through all the volumes of the journal Journal of the History of Ideas at a nearby university library. Incidentally, Whitmore's paper is where I learned about speculations of James Fitzjames Stephen that are given above. The second new item, [B], is an excerpt from an essay by Augustus De Morgan that I read this last weekend. De Morgan's essay is item [15] in my answer to the History of Science and Math StackExchange question Did Galileo's writings on infinity influence Cantor?, and his essay is also mentioned in item [8]. I've come across references to De Morgan's essay from time to time over the years, but I've never read it because I never bothered trying to look it up in a university library. However, when I found to my surprise (but I really shouldn't have been surprised) that a digital copy of the essay was freely available on the internet when I searched for it about a week ago, I made a print copy, which I then read through when I had some time (this last weekend).

[A] Charles Edward Whitmore (1887-1970), Mill and mathematics: An historical note, Journal of the History of Ideas 6 #1 (January 1945), 109-112. MR 6,141n; Zbl 60.01622

(first paragraph of the paper, on p. 109) In various philosophical works one encounters the statement that J. S. Mill somewhere asserted that two and two might conceivably make five. Thus, Professor Lewis says$^1$ that Mill "asked us to suppose a demon sufficiently powerful and maleficent so that every time two things were brought together with two other things, this demon should always introduce a fifth"; but he gives no specific reference. {{footnote: $^1$C. I. Lewis, Mind and the World Order (1929), 250.}} C. S. Peirce$^2$ puts it in the form, "when two things were put together a third should spring up," calling it a doctrine usually attributed to Mill. {{footnote: $^2$Collected Papers, IV, 91 (dated 1893). The editors supply a reference to Logic, II, vi, 3.}} Albert Thibaudet$^3$ ascribes to "a Scottish philosopher cited by Mill" the doctrine that the addition of two quantities might lead to the production of a third. {{footnote: $^3$Introduction to Les Idées de Charles Maurras (1920), 7.}} Again, Professor Laird remarks$^4$ that "Mill suggested, we remember, that two and two might not make four in some remote part of the stellar universe," referring to Logic III, xxi, 4 and II, vi, 2. {{footnote: $^4$John Laird, Knowledge, Belief, and Opinion (1930), 238.}} These instances, somewhat casually collected, suggest that there is some confusion in the situation.

(from pp. 109-111) Moreover, the notion that two and two should ["could" intended?] make five is entirely opposed to the general doctrine of the Logic. $[\cdots]$ Nevertheless, though these views stand in the final edition of the Logic, it is true that Mill did, in the interval, contrive to disallow them. After reading through the works of Sir William Hamilton three times, he delivered himself of a massive Examination of that philosopher, in the course of which he reverses his position--but at the suggestion of another thinker. In chapter VI he falls back on the inseparable associations generated by uniform experience as compelling us to conceive two and two as four, so that "we should probably have no difficulty in putting together the two ideas supposed to be incompatible, if our experience had not first inseparably associated one of them with the contradictory of the other." To this he adds, "That the reverse of the most familiar principles of arithmetic and geometry might have been made conceivable even to our present mental faculties, if those faculties had coexisted with a totally different constitution of external nature, is ingeniously shown in the concluding paper of a recent volume, anonymous, but of known authorship, Essays, by a Barrister." The author of the work in question was James Fitzjames Stephen, who in 1862 had brought together various papers which had appeared in Saturday Review during some three previous years. Some of them dealt with philosophy, and it is from a review of Mansel's Metaphysics that Mill proceeds to quote in support of his new doctrine $[\cdots]$

Note: On p. 111 Whitmore argues against Mill's and Stephen's empirical viewpoint of "two plus two equals four". Whitmore's arguments are not very convincing to me.

(from p. 112) Mill, then, did not originate the idea, but adopted it from Stephen, in the form that two and two might make five to our present faculties, if external nature were differently constituted. He did not assign it to some remote part of the universe, nor did he call in the activity of some maleficent demon; neither did he say that one and one might make three. He did not explore its implications, or inquire how it might be reconciled with what he had said in other places; but at least he is entitled to a definite statement of what he did say. I confess that I am somewhat puzzled at the different forms in which it has been quoted, and at the irrelevant details which have been added.

[B] Augustus De Morgan (1806-1871), On infinity; and on the sign of equality, Transactions of the Cambridge Philosophical Society 11 Part I (1871), 145-189.

Published separately as a booklet by Cambridge University Press in 1865 (same title; i + 45 pages). The following excerpt is from the version published in 1865.

(footnote 1 on p. 14) We are apt to pronounce that the admirable pre-established harmony which exists between the subjective and objective is a necessary property of mind. It may, or may not, be so. Can we not grant to omnipotence the power to fashion a mind of which the primary counting is by twos, $0,$ $2,$ $4,$ $6,$ &c.; a mind which always finds its first indicative notion in this and that, and only with effort separates this from that. I cannot invent the fundamental forms of language for this mind, and so am obliged to make it contradict its own nature by using our terms. The attempt to think of such things helps towards the habit of distinguishing the subjective and objective.

Note: Those interested in such speculations will also want to look at De Morgan's lengthy footnote on p. 20.

(ADDED 6 YEARS LATER) I recently read Ian Stewart's 2006 book Letters to a Young Mathematician and in this book there is a passage (see below) that I think is worth including here.

(from pp. 30-31) I think human math is more closely linked to our particular physiology, experiences, and psychological preferences than we imagine. It is parochial, not universal. Geometry's points and lines may seem the natural basis for a theory of shape, but they are also the features into which our visual system happens to dissect the world. An alien visual system might find light and shade primary, or motion and stasis, or frequency of vibration. An alien brain might find smell, or embarrassment, but not shape, to be fundamental to its perception of the world. And while discrete numbers like $1,$ $2,$ $3,$ seem universal to us, they trace back to our tendency to assemble similar things, such as sheep, and consider them property: has one of my sheep been stolen? Arithmetic seems to have originated through two things: the timing of the seasons and commerce. But what of the blimp creatures of distant Poseidon, a hypothetical gas giant like Jupiter, whose world is a constant flux of turbulent winds, and who have no sense of individual ownership? Before they could count up to three, whatever they were counting would have blown away on the ammonia breeze. They would, however, have a far better understanding than we do of the math of turbulent fluid flow.

Answer 3

I personally would define the symbol $2$ as $1+1$.

However, depending on what you assume, this does not ensure $2\ne 0$. What do people usually assume? Often, mathematicians work with the notion of a commutative ring. Think of the integer numbers in the following definition: A commutative ring is a mathematical structure with an addition operator $+$ and a multiplication operator $\times$, containing elements $1$ and $0$ subject to the conditions that

• $a + b = b + a$ and $a\times b = b\times a$
• $a + (b + c) = (a + b) + c$ and $a \times (b \times c) = (a \times b)\times c$
• $a + 0 = a$ and $a\times 1 = a$
• For each $a$ there exists $b$ with $a+b=0$, we write $b=-a$
• We have $a\times(b+c)=(a\times b) + (a\times c)$

Now the integer numbers clearly satisfy these conditions, and we all have a good idea what $2$ means there. However, consider $\{0,1\}=:\mathbb F_2$. It also satisfies the conditions, if we perform addition and subtraction as usual, but decree that $1+1=0$. Now you have $2=0$ inside $\mathbb F_2$.

However, note that nothing stops you from limiting yourself to mathematical structures where $1+1\ne 0$. In fact, quite often, one limits himself to structures where $1+\cdots+1\ne 0$ no matter how many times you add $1$ to itself.

In the end, it's all just a matter of definitions. But truly, I would always say $2=1+1$ simply because the symbol $2$ should reasonably be defined as that.

Answer 4

Here's an argument to consider. Take the following three propositions:

(A) 1 + 1 = 2

(B) If there is exactly one $F$ and exactly one $G$ (and no $F$ is $G$), then there are exactly two things which are $F$-or-$G$.

(C) $(\exists x)(Fx \land \forall y(Fy \to y = x)$, $(\exists x)(Gx \land \forall y(Gy \to y = x) \vdash$ $\quad\quad\quad(\exists x)(\exists y)(\neg x = y \land (Fx \lor Gx) \land (Fy \lor Gy) \land (\forall z)(Fz \lor Gz \to (z = x \lor z = y))$

Now, it is a vexed question what exactly the relation is between (A) and (B). But it is very plausible to say that if (A) is indeed interpreted as a statement of ordinary schoolroom arithmetic (and not as a mere de-intepreted string of formal symbols), then so understood, (A) is true iff, for any $F$, $G$, (B) is true. But (B), with numerical quantifiers, can be regimented as (C), with ordinary quantifiers. And (C) is a straight logical theorem of first-order logic.

So (C) is true as a matter of logical necessity, hence (B) is. And it looks a priori that (A)'s truth necessarily goes with the truth of (B) for any $F$, $G$. So it follows that (A), understood the ordinary way, is indeed necessarily true. too.