Given my understanding of epistemic and necessary truths it seems plausible that I can construct epistemic truths using only necessary ones, which feels contradictory.
Eg 1 + 1 = 2 is a necessary truth
But 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 10 is epistemic. It could very easily be wrong if I have miscounted the number of 1s.
This seems to suggest that necessary truths are just “simple to check” and that sufficiently complex necessary truths become epistemic because of a failure to check an operation.
Similarly “there are 180 degrees on the inside of a triangle” is only necessarily true in spaces such as R2. It might look necessarily true everywhere but it’s not on the sphere. So what looks like a necessary truth actually an epistemic one.
Is it a necessary non-epistemic truth? After all, it has a very lengthy partial proof in Principia Mathematica, and maybe they got something wrong. Perhaps you should check?
But then maybe you’re not using a formal system to prove it, but just taking it as an axiom or maybe as a definition of what “2” means using other symbols with pre-existing meanings. But then if I define the term “blerg” to mean “a breakfast product with non-obvious composition”, is that definition in itself a necessary truth?
Obviously if you mean “if you take one object and then take another object, you now have two objects” then that’s a contingent proposition that requires evidence. It probably depends upon what sorts of things you mean by “objects” too, so we can rule that one out.
Or maybe “necessary non-epistemic truth” means a proposition that you can “grok in fullness” and just directly see that it is true as a single mental operation? Though, isn’t that subjective and also epistemic? Don’t you have to check to be sure that it is one? Was it a necessary non-epistemic truth for you when you were young enough to have trouble with the concept of counting?
So in the end I’m not really sure exactly what you mean by a necessary truth that doesn’t need any checking. Maybe it’s not even a coherent concept.
What do you mean by “necessary truth” and “epistemic truth”? I’m sorta confused about what you are asking.
I can be uncertain about the 1000th digit of pi. That doesn’t make the digit being 9 any less valid. (Perhaps what you mean by necessary?) Put another way, the 1000th digit of pi is “necessarily” 9, but my knowledge of this fact is “epistemic”. Does this help?
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.
Given my understanding of epistemic and necessary truths it seems plausible that I can construct epistemic truths using only necessary ones, which feels contradictory.
Eg 1 + 1 = 2 is a necessary truth
But 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 10 is epistemic. It could very easily be wrong if I have miscounted the number of 1s.
This seems to suggest that necessary truths are just “simple to check” and that sufficiently complex necessary truths become epistemic because of a failure to check an operation.
Similarly “there are 180 degrees on the inside of a triangle” is only necessarily true in spaces such as R2. It might look necessarily true everywhere but it’s not on the sphere. So what looks like a necessary truth actually an epistemic one.
What am I getting wrong?
Is it a necessary non-epistemic truth? After all, it has a very lengthy partial proof in Principia Mathematica, and maybe they got something wrong. Perhaps you should check?
But then maybe you’re not using a formal system to prove it, but just taking it as an axiom or maybe as a definition of what “2” means using other symbols with pre-existing meanings. But then if I define the term “blerg” to mean “a breakfast product with non-obvious composition”, is that definition in itself a necessary truth?
Obviously if you mean “if you take one object and then take another object, you now have two objects” then that’s a contingent proposition that requires evidence. It probably depends upon what sorts of things you mean by “objects” too, so we can rule that one out.
Or maybe “necessary non-epistemic truth” means a proposition that you can “grok in fullness” and just directly see that it is true as a single mental operation? Though, isn’t that subjective and also epistemic? Don’t you have to check to be sure that it is one? Was it a necessary non-epistemic truth for you when you were young enough to have trouble with the concept of counting?
So in the end I’m not really sure exactly what you mean by a necessary truth that doesn’t need any checking. Maybe it’s not even a coherent concept.
What do you mean by “necessary truth” and “epistemic truth”? I’m sorta confused about what you are asking.
I can be uncertain about the 1000th digit of pi. That doesn’t make the digit being 9 any less valid. (Perhaps what you mean by necessary?) Put another way, the 1000th digit of pi is “necessarily” 9, but my knowledge of this fact is “epistemic”. Does this help?
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.