I asked the same question at math.stackexchange but I thought that the answers here could be quite different.

I hope I am not breaking protocol/etiquette by doing that, if I am then I apologize.

If I am breaking protocol/etiquette then is there a better way for me to ask this question to both stackexchange sections.

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    You've tagged this philosophy of math. Maybe it's just me, but I think this turns out to be more about philosophy language and intension. What do you mean by the two assertions? Can you clarify whether you think the conundrum is merely linguistic or hinges or something distinctly meant by the two expressions? – virmaior Feb 21 '16 at 13:53
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    "Is" and "equals" are in this context synonyms... I don't see the philosophical question here. – James Kingsbery Feb 21 '16 at 20:05

We may say that the two assertions, i.e. two different speech acts or two different written expression (grammatically well-formed string of symbols) express the same mathematical fact (or thought or content).

See also propositions and Sense and reference for Frege's Philosophy of language..


The difference is that the first uses "equals" which in logic and mathematics is the identity predicate, whereas the second uses "is" which is an English word with a variety of uses, including ascribing a property, stating a relationship, expressing a membership or subclass relation, introducing a definition, or just acting as an auxiliary verb. There is another question about "is" here with a lot of useful answers.

In the context of the sentence "2 + 2 is 4", it is only feasible to interpret the "is" to mean equality, so in practice the two mean the same thing here. We might express that thing to be that the referent of the expression "s(s(0)) + s(s(0))" is numerically identical with the referent of the expression "s(s(s(s(0))))".


We can say "Plato is a man" which in set theory means that Plato is an element in the set of Men. This doesn't imply that A man is Plato.

One axiom of equality is that if A=B then B=A

Plato = A man implies A man = Plato, however that is not true

The axiom applies to real numbers but it doesn't apply to nouns. So the short answer is: it depends which axioms of equality you take.


In the most technical sense of the phrasings, they are different relationships. Equality and identity are typically not assumed to be the same until one adds an axiom that states that they are.

Consider a hypothetical example involving baskets and gold. We each have one basket, and they have an equal amount of gold in them. One could say the baskets are "equal," and in all but the most extreme meanings of that word, we would agree that indeed, both baskets are equal. Now I produce a new bar of gold and announce that I am going to put it in one of the baskets. Do you care if I put it in your basket or my own?

If equality and identity are the same concept, then you should not care which basket it goes in, using the logic "because both baskets are equal, they are identical so I should not care where the extra bar of gold goes." Basic intuition screams at you that this is not the case. You do care which basket the gold goes in, because you would like to take more gold home with you. Thus, basic intuition suggests that equality and identity are fundamentally different concepts.

This shows up in many places. In programming, two references to objects that are equal to eachother is treated very differently than two references to the same object. Also, intriguingly, in math you do not always have to define an equality function for the system you are working with. Typically you do choose to define one, because they're convenient, but you don't have to. In such case, you might have an identity relationship which does not imply equality (A is not equal to itself!) because you did not define equality in the first place.

All of that technicality aside, one does have to consider the colloquial usage of the phrasings. It is generally assumed that, if someone says "2 + 2 is 4," they really meant the same thing as "2 + 2 equals 4." People often choose to lose precision in exchange for making it easier to speak. In everyday life, if someone told me "2 + 2 is 4," that is the assumption I would use. However, if I was deep in a philosophical discussion about mathematics, including the topics of equality and identity, I may not make that assumption. I may instead ask them to clarify whether they intended to use the equality relationship or the identity relationship.

One example of where this shows up is predicate logic with the symbol, which is one of the most common symbol used for "implication." In predicate logic, implication is considered a meta-language operator. It is not part of predicate logic, but rather it is the symbol which describes an intuitive concept that suggests one can substitute one predicate for another, according to a set of rules. It is unique because it is the only symbol in predicate logic whose semantics are not described using predicate logic itself. Thus, if I was having a discussion about predicate logic, and somebody used the verb "is" to link two concepts, I may have to ask them to clarify whether they are referring to the equality concept within the predicate logic, or if they are referring to the implication concept outside of predicate logic. If it is the former, it is understood that the validity of their statement can be proven using predicate logic. If it is the latter, then it may be a more nuanced leap that needs more discussion.

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