Given a flavor of nominalism which denies that simple sentences and existential quantifiers referring to mathematical objects are literally true (pretense theory, fictionalism, figuralism, etc.), suppose there is a circle, C. According to my understanding, either of the propositions "the radius of C is 3 feet" or "the area of C is 5 square feet" could be literally true, since "3" and "5" only serve to specify the cardinality of "feet" and "square feet," and are not purported to imply the real existence of "3" or "5."

Now, when we combine these two propositions, does an actual contradiction technically arise, or would we move into the realm of pretense/fiction/figurativ-ity when stating the two propositions in terms of the same units (likely square feet) since the units would be canceled out? We would assess the compatibility of the propositions by setting them equal to one another:

area = pi * radius^2

5 square feet = pi * (3 feet)^2

5 square feet = pi * 9 square feet

5 = pi * 9

5 = 28.2743... (false)

My question is, since the units were canceled out by the time we arrived at an explicit contradiction (we might say they were paraphrased away when we thought about the propositions in their most reduced form), did we lose the ability to say that these propositions are contradictory on these various nominalist views?

  • Having derived a contradiction from the two premises, we have to conclude that at least one of them must be false, which indeed is: if the circle C has radius 3, then its area cannot be equal to 5. – Mauro ALLEGRANZA May 15 '16 at 10:40
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    "Literal truth" is not a useful notion under nominalism since nothing is "literally true". So arguments involving "literal truth" are completely irrelevant to nominalistic mathematics. Fictionalists for example use "true-according-to" instead to reproduce it. – Conifold May 15 '16 at 19:33

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