The Proof is Trivial!

Problem 200***

Evaluate $\displaystyle\int_0^1 \ln(t) t^2 dt$

Isn't that because mathematical induction is actually deduction haha?

Interestingly, I'm not sure if that is a solution the Problem of Induction: you still need to find a way to justify your confidence level, do you not? And how are you going to do that, if not by reasoning inductively in the first place? And for this you will need another confidence interval. And so on, ad infinitum.

There's a problem very closely linked to the problem of induction that I've always found fascinating and extremely frustrating. Again, I think it is a pretty old di/tri/multilemma, but it's always fun to think about!

Any valid argument must rely on either a circular chain of reasons, an infinite chain of reasons or on reasons that are not themselves justified.
An argument cannot be justified by a circular chain of reasons.

An argument cannot be justified by an infinite chain of reasons.

An argument cannot be justified if it relies upon unjustified assumptions.

Thus, no valid argument can be justified.

Solution 200

$\displaystyle\int_{0}^{1} \ln t \cdot t^{2} \ dt \overset{ \text{IBP}}= \displaystyle\underbrace{ \dfrac{1}{3} t^{3} \ln t \bigg|_{0}^{1}}_{*} - \dfrac{1}{3} \displaystyle\int_{0}^{1} t^{2} \cdot \dfrac{1}{t} \ dt = - \dfrac{1}{9}$

Why is this ***? I presume you want justification of the $*$ limit...

$\dfrac{1}{3} t^{3} \ln t \bigg|_{0}^{1} = - \dfrac{1}{3} \left( \displaystyle\lim_{t \to 0} t^{3} \ln t \right)$

$\overset{ \text{L'Hopital's rule}}= \dfrac{1}{3} \cdot \left( \displaystyle\lim_{t \to 0} \dfrac{t^{3}}{3} \right) = 0$

$\therefore \displaystyle\int_{0}^{1} \ln t \cdot t^{2} \ dt = - \dfrac{1}{9}$

Problem 195*/**

Prove that there exists infinitely many arithmetic progressions of 3 distinct perfect squares.

Problem 196**/***

Prove that there exists no arithmetic progressions of 4 distinct perfect squares.

Problem 197*/**

For a non-negative integer n, is $5^{5^{n+1}}+5^{5^{n}}+1$ ever prime? Prove your assertions.

Problem 198**

Let the real numbers a,b,c,d satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12$.

Prove that $36 \le 4(a^3+b^3+c^3+d^3)-(a^4+b^4+c^4+d^4) \le 48$

Problem 199***

Prove that $\displaystyle \int_0^1 t^x (1-t)^y dt = \frac{\Gamma(x+1) \Gamma(y+1)}{\Gamma(x+y+2)}$

Hint:

Problem 200***

Evaluate $\displaystyle\int_0^1 \ln(t) t^2 dt$

Actually I found it in a list of problems to practice using the Gamma Function and results such as that obtained in 199 and so hadn't really stopped to consider how it might be approached outside of this context (the advanced method is still worth doing though if you're up for it!) Your solution is far better

You say 'no valid argument' can be justified. Does that mean your argument is invalid as you have justified it?

No; not quite, I don't think. The argument I presented is clearly valid in that its premises entail its conclusion. Whether or not it is justified is another matter! If the conclusion is justified, then it is correct and the conclusion cannot be justified or we have a contradiction. So the conclusion cannot be justified. If the conclusion is not justified, then everything is fine i.e. there is no contradiction. So we must conclude that the conclusion is unjustified (though not necessarily false) as it must either be justified or unjustified and we have shown that it is not justified.

So we need to look at where the problem might lie, and I think the answer is clear: the argument rests upon unjustified assumptions (axioms) as, I think, do most/all arguments. And so it boils down to the idea behind the problem in the first place: if an argument rests on assumptions that are not inferentially justified, then how can that argument ultimately be justified?

Now I think an axiom (one of the initial assumptions we don't justify) can be of two types; it can be a defining property of something (such as with most axioms in mathematics, I think), or it can be "self-evidently true". And I think, aside from tautologies that give no new information, the idea of anything about reality being "self-evidently" true is unrealistic. We could always be wrong about what is "self-evidently" true.


But hey, I'm just a young man trying to sound cleverer than he is I'm not a philosopher; nor do I have a mathematics degree. I could be entirely wrong

No a confidence level constructed correctly is deductively justified, for example through interpretations of probability or statistical inference, which are sound albeit open to disputes of opinion.

That depends on the nature of your argument. For example Axiom of Choice => Zorn's Lemma => Well-Ordering Theorem => Axiom of Choice is a well-known circular set of results that stand independently from traditional foundations of set theory, and is unprovable from them unless you assume one of the equivalent (or stronger) forms. But that just reduces to the matter of what you pick as your axioms.



Yes they can.



Again, depends on the argument. A vacuous or tautological argument can have false assumptions.



Essentially the whole thing is ignoring the key foundation of deduction that justification is only relative to some set of axioms - within that axiomatic system the deductions could be perfectly sound, albeit perhaps open to dispute of matters of opinion (choice of axioms). In a carefully constructed logical system even infinite proofs can be shown to sound, and there are plenty of philosophical justifications for why one may like to choose to adopt that instead of the more classical systems we are used to that does not allow infinite proof.

Tl;dr: What you said appears to be a trilemma only because you are confining yourself to classic logic, and even then it's a false one since it ignores the fact that arguments are justified only relative to some axioms.

The second part can be done in three lines by using only value of Hurwitz's zeta function.

I think I got a little confused in my last post, but I'll try to explain my thoughts again now. It may well be that I just don't understand, though.

Say we've measured the stress levels of managers and floor staff in a sample of shops and found that, in general, managers are more stressed than floor staff. We can do our stats to compute the probability that the difference in stress measurements between managers and floor staff is merely due to random chance. But, without any inductive inferences, we can't make any judgement about what did cause this difference: we can only decide whether it was likely to be due to random chance or a real variable difference. If it is likely to be due to a real variable, the stats still hasn't told anything about that variable beyond the fact that it isn't random chance.

With regard to the icebergs, wouldn't our confidence interval necessarily rely on assumptions and inductive inferences? We may well have measured 1 000 000 icebergs and done our stats to find that we can be 99.7% certain that all icebergs are cold, for example... But what if there is a different type of underground iceberg that forms under pressure - and at higher temperatures - that we didn't know about then and is subsequently discovered? The stats may well have told us that the probability was 99.7%, but, without sounding like an excitable philosophy freshman with an obsession with Descartes and his method of doubt, were we really in a position to make that call when we had no way of knowing what information and data we were missing regarding icebergs in the first place? Our statistical calculation didn't (and couldn't have) taken this into account, so how can we ever have really been so sure in the first place?

I think that is exactly the point of the problem, aside from a difference in usage of the term "justification". Any argument we make necessarily relies upon some assumptions; axioms, doesn't it? So whether or not an argument is valid is entirely relative to these axioms.

All cheeses are purple.
All iPods are cheese.
Therefore all iPods are purple.

This is a valid argument and is "true" relative to the assumptions that all cheeses are purple and all ipods are cheese... But wouldn't it be absurd to claim that the conclusion was ultimately justified (that is, it is appropriate to believe it)? If any argument we make relies upon something that we don't check to be true - we just assume it - then how can we claim to be justified when saying that any proposition is true or false? Our argument for its truth or falsity is necessarily based upon premises that we haven't verified.

If we have a reason to believe that one axiom is "better" or "truer" than another, then aren't we really justifying it? If so, then it isn't really an axiom, is it?

Problem 195*/**

Prove that there exists infinitely many arithmetic progressions of 3 distinct perfect squares.

Problem 201:

A stick of unit length is bent at a point along its length, with each such point being equally likely. The stick is then placed on a horizontal surface so that it forms a right-angled triangle, with the shorter edge of the stick being perpendicular to the ground. The angle of elevation of the longer edge of the stick with the ground is $\alpha$. Prove that the expected value of $\alpha$ is given by:

$\dfrac{4- \pi }{4}$

Problem 201:

A stick of unit length is bent at a point along its length, with each such point being equally likely. The stick is then placed on a horizontal surface so that it forms a right-angled triangle, with the shorter edge of the stick being perpendicular to the ground. The angle of elevation of the longer edge of the stick with the ground is $\alpha$. Prove that the expected value of $\alpha$ is given by:

$\dfrac{4- \pi }{4}$

First question - Is this a */**/*** problem?

Second question - Is this stick being dropped from the Eiffel Tower?

I didn't expect this to be so pure

First question - Is this a */**/*** problem?

Second question - Is this stick being dropped from the Eiffel Tower?