Just a note, in conventional philosophical terminology you would say:
Eg 1 + 1 = 2 is an epistemic necessity
But 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 10 is an epistemic contingency.
One way to interpret this is to say that your degree of belief in the first equation is 1, while your degree of belief in the second equation is neither 1 nor 0.
Another way to interpret it is to say that the first is “subjectively entailed” by your evidence (your visual impression of the formula), but not the latter, nor is its negation.A subjectively entails B iff P(B∣A)=1, where P is a probability function that describes your beliefs.
In general, philosophers distinguish several kinds of possibility (“modality”).
Epistemic modality is discussed above. The first equation seems epistemically necessary, the second epistemically continent.
With metaphysical modality (which roughly covers possibility in the widest natural sense of the term “possible”), both equations are necessary, if they are true. True mathematical statements are generally considered necessary, except perhaps for some more esoteric “made-up” math, e.g. more questionable large cardinal axioms. This type is usually implied when the type of modality isn’t specified.
With logical modality, both equations are logically contingent, because they are not logical tautologies. They instead depend on some non-logical assumptions like the Peano axioms. (But if logicism is true, both are actually disguised tautologies and therefore logically necessary.)
Nomological (physical) modality: The laws of physics don’t appear to allow them to be false, so both are nomologically necessary.
Analytic/synthetic statements: Both equations are usually considered true in virtue of their meaning only, which would make them analytic (this is basically “semantic necessity”). For synthetic statements their meaning would not be sufficient to determine their truth value. (Though Kant, who came up with this distinction, argues that arithmetic statements are synthetic, although synthetic a priori, i.e. not requiring empirical evidence.)
Anyway, my opinion on this is that “1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 10” is interpreted as the statement “this bunch of ones [referring to screen] added together equal 10″ which has the same truth value, but not the same meaning. The second meaning would be compatible with slightly more or fewer ones on screen than there actually are on screen, which would make the interpretation compatible with a similar false formula which is different from actual one. The interpretation appears to be synthetic, while the original formula is analytic.
This is similar to how the expression “the Riemann hypothesis” is not synonymous to the Riemann hypothesis, since the former just refers to a statement instead of expressing it directly. You could believe “the Riemann hypothesis is true” without knowing the hypothesis itself. You could just mean “this bunch of mathematical notation expresses a true statement” or “the conjecture commonly referred to as ‘Riemann hypothesis’ is true”. This belief expresses a synthetic statement, because it refers to external facts about what type of statement mathematicians happen to refer to exactly, which “could have been” (metaphysical possibility) a different one, and so could have had a different truth value.
Basically, for more complex statements we implicitly use indexicals (“this formula there”) because we can’t grasp it at once, resulting in a synthetic statement. When we make a math mistake and think something to be false that isn’t, we don’t actually believe some true analytic statement to be false, we only believe a true synthetic statement to be false.
Just a note, in conventional philosophical terminology you would say:
One way to interpret this is to say that your degree of belief in the first equation is 1, while your degree of belief in the second equation is neither 1 nor 0.
Another way to interpret it is to say that the first is “subjectively entailed” by your evidence (your visual impression of the formula), but not the latter, nor is its negation.A subjectively entails B iff P(B∣A)=1, where P is a probability function that describes your beliefs.
In general, philosophers distinguish several kinds of possibility (“modality”).
Epistemic modality is discussed above. The first equation seems epistemically necessary, the second epistemically continent.
With metaphysical modality (which roughly covers possibility in the widest natural sense of the term “possible”), both equations are necessary, if they are true. True mathematical statements are generally considered necessary, except perhaps for some more esoteric “made-up” math, e.g. more questionable large cardinal axioms. This type is usually implied when the type of modality isn’t specified.
With logical modality, both equations are logically contingent, because they are not logical tautologies. They instead depend on some non-logical assumptions like the Peano axioms. (But if logicism is true, both are actually disguised tautologies and therefore logically necessary.)
Nomological (physical) modality: The laws of physics don’t appear to allow them to be false, so both are nomologically necessary.
Analytic/synthetic statements: Both equations are usually considered true in virtue of their meaning only, which would make them analytic (this is basically “semantic necessity”). For synthetic statements their meaning would not be sufficient to determine their truth value. (Though Kant, who came up with this distinction, argues that arithmetic statements are synthetic, although synthetic a priori, i.e. not requiring empirical evidence.)
Anyway, my opinion on this is that “1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 10” is interpreted as the statement “this bunch of ones [referring to screen] added together equal 10″ which has the same truth value, but not the same meaning. The second meaning would be compatible with slightly more or fewer ones on screen than there actually are on screen, which would make the interpretation compatible with a similar false formula which is different from actual one. The interpretation appears to be synthetic, while the original formula is analytic.
This is similar to how the expression “the Riemann hypothesis” is not synonymous to the Riemann hypothesis, since the former just refers to a statement instead of expressing it directly. You could believe “the Riemann hypothesis is true” without knowing the hypothesis itself. You could just mean “this bunch of mathematical notation expresses a true statement” or “the conjecture commonly referred to as ‘Riemann hypothesis’ is true”. This belief expresses a synthetic statement, because it refers to external facts about what type of statement mathematicians happen to refer to exactly, which “could have been” (metaphysical possibility) a different one, and so could have had a different truth value.
Basically, for more complex statements we implicitly use indexicals (“this formula there”) because we can’t grasp it at once, resulting in a synthetic statement. When we make a math mistake and think something to be false that isn’t, we don’t actually believe some true analytic statement to be false, we only believe a true synthetic statement to be false.