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My professor of Epistemological Basis of Modern Science discipline was questioning about what we consider knowledge and what makes us believe or not in it's reliability.

To test us, he asked us to write down our justifications for why do we accept as true that 2 plus 2 is equal to 4. Everybody, including me, answered that we believe in it because we can prove it, like, I can take 2 beans and more 2 beans and in the end I will have 4 beans. Although the professor told us: "And if all the beans in the universe disappear", and of course he can extend it to any object we choose to make the proof. What he was trying to show us is that the logical-mathematical universe is independent of our universe.

Although I was pretty delighted with this question and I want to go deeper. I already searched about Peano axioms and Zermelo-Fraenkel axioms although I think the answer that I am looking for can't be explained by an axiom.

It is a complicated question for me, very confusing, but try to understand, what I want is the background process, the gears of addition, like, you can say that a+0=a and then say a+1 = a+S(0) = S(a+0) = S(a). Although it doesn't show what the addition operation itself is. Can addition be represented graphically? Like rows that rotates, or lines that join?

Summarizing, I think my question is: How can I understand addition, not only learn how to do it, not just reproduce what teachers had taught to me like a machine. How can I make a mental construct of this mathematical operation?

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closed as off topic by Asaf Karagila, Shuhao Cao, Erick Wong, martini, Start wearing purple Jun 7 '13 at 23:16

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If it can't be explained by the axioms, perhaps it's not really a mathematical question, and should be asked on/migrated to philosophy.SE? – Asaf Karagila Jun 7 '13 at 22:37
Is there a difference between "I can take 2 beans and 2 more beans and in the end I will have 4 beans", and "2+2=4"? The second statement is a theorem. We have some definitions, and some axioms, and using them we can prove this. That "2+2=4" is true essentially means that from a standard set of axioms using standard rules of inference we can produce a formal proof of this statement. The first one seems to be an observation about the real world. Observations about the real world are empirical facts, they cannot be proved. Is the question about the relation between these 2 statements? – Andrés E. Caicedo Jun 7 '13 at 22:38
Mathematics did just fine for a long time without induction or PA. – vadim123 Jun 7 '13 at 22:51
It seems to me that, if all the beans in the universe were to vanish, the fact that $2+2=4$ would survive unscathed, but the familiar connection between addition and bean-counting would have to be discarded or at least revised. In other words, mathematics is not about beans but about abstract patterns that may or may not have physical realizations. Patterns can remain even if their physical realizations get ruined. – Andreas Blass Jun 7 '13 at 23:18
When the beans disappear, we can still imagine the beans. Or any other objects. And when our imagination disappears? Well, then we still can count our thoughts. And if the thoughts disappear as well? Well, then there's nobody left who could ask the question. – celtschk Jun 7 '13 at 23:30
up vote 4 down vote accepted

I've always liked this approach, that a naming precedes a counting.


enter image description here


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Thanks, your answer combined to the comments on my question clarified my mind about this matter. – Rodrigo Martins Jun 8 '13 at 0:27

Take the set $2=\{0,1\}$ (this is indeed $2$, nothing more, nothing less) and consider the set $A:=2\times \{0\}\cup 2\times \{1\}$ (two disjoint copies of 2). The number of elements (the cardinality) of such a set is (one of the possibility for) what we mean by $2+2$ and such a set has exactly four elements, i.e. there's a bijection between $A$ and $4=\{0,1,2,3\}$. This is why $2+2=4$, I'd say.

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I find that much less persuasive than the argument about the beans. – MJD Jun 7 '13 at 23:06

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