PrepTest 101, Section 2, Question 24
No mathematical proposition can be proven true by(15 more words not shown)
The conclusion follows logically if(7 more words not shown)
- AOnly propositions that(10 more words not shown)
- BObservation alone cannot(9 more words not shown)
- CIf a proposition(14 more words not shown)
- DKnowing a proposition(14 more words not shown)
- EKnowing a proposition(9 more words not shown)
Reading the argument
What about a proof that doesn't rely on observation?
We are told mathematical claims can't be proved true by observation. The author jumps from that limitation to saying we can't know any of them to be true. But perhaps reasoning or calculation could establish a mathematical claim in another way. Ruling out one route isn't the same as ruling out every route.
What we’re looking for
To make the conclusion follow, we need to close that opening. My prediction is that knowing a proposition to be true requires proof by observation. If that kind of proof is required, and the premise says mathematical propositions can't have it, then they can't qualify as things we know to be true. We don't need to settle a real debate about how mathematical knowledge works; we need a rule that completes this argument.
Working the answers
- A
This requires a proposition to be provable before we can know it is true. But it doesn't limit the kind of proof. A mathematical proposition could be proved through some method other than observation, satisfying A while remaining outside the method ruled out in the premise. A leaves the exact opening we identified.
- B
This expands the limitation on observation to all propositions, but it still doesn't rule out other methods. Perhaps observation combined with reasoning could establish something, or perhaps reasoning alone could. Saying observation alone isn't enough does not establish that knowing mathematical truths is impossible.
- C
This tells us what follows if observation can prove a proposition: it can then be known to be true. But our premise says mathematical propositions cannot be proved that way. A rule about cases where observation succeeds doesn't tell us that every other route to knowledge fails.
- D
Read “only if” carefully. D says that when knowing a proposition is impossible, proof by observation must also be unavailable. We need to get from unavailable observational proof to impossible knowledge, and D doesn't authorize that step.
For example, a mathematical claim might be knowable through reasoning even though observation cannot prove it. D doesn't rule that situation out: it hasn't said that lacking observational proof always makes knowledge impossible. E does supply the requirement that blocks this possibility.
- ECorrect
Knowing a proposition to be true requires proving it true by observation. The premise says no mathematical proposition can receive that kind of proof. Each therefore lacks something E makes required for knowledge, so the conclusion follows: no mathematical proposition can be known to be true.
E isn't just saying observation is one useful method. It makes observational proof a requirement. That's why the inability to use this method now rules out knowledge, rather than leaving other ways open.
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