I'm a mathematician looking at some of the work of Leonhard Euler on the "pentagonal number theorem". My question is about how we can know some statement is true. Euler had found this theorem in the early 1740s, and said things like "I believed I have concluded it by a legitimate induction, but at the same time I haven't been able to find a demonstration" (my translation), and that it is "true even without being demonstrated" (vraies sans etre demontrees). This got me thinking that "knowing" something is not really a mathematical question. A proof lets us know a statement is true because we can work through the proof. But a mathematical statement is true whether we know it or not, and if you tell me you know that a statement is true, and then in fact someone later proves it, I can't show mathematically that you didn't know it all along. This isn't something I have thought about much before, and my question is are there any papers or books that give some ideas about this that would be approachable by...

This problem has been nagging me forever. If "objective reality" is simply a consensus between experiencing subjects, then on what grounds can we claim to know or understand anything? How can we be so sure that - for example - our scientific knowledge is accurate? Is it just because there is greater consensus in established academic fields like physics or biology? What about the people our society labels as insane? Is their interpretation of reality wrong simply because there is less consensus about it?

How do we know that plants and similar don't feel pain? As far as I see it plants just don't act like they're in pain, but that doesn't mean they aren't. They could just be very stoic about it.

John Carey has written a book called "What good are the arts?" His central idea is that our evaluation of the visual arts and music is completely subjective and relativistic. Art and art creation are seen as an important part of being human but no one can make a case for a work being of higher value because this is just opinion. Fine. However, he goes on to argue the case for the higher value of literature. Predicting the obvious objections one might have after his previous relativist argument he says: "let me emphasize that all the judgements made in this part including the judgment of what 'literature' is are inevitably subjective". Here and in live debates he has stated this as a means of getting himself off his own hook. So my question is, surely there is some contradiction involved in arguing a position while at the same time stating that it is just subjective? Aren't we trying to lay claim to some objective truth as soon as we begin arguing?

Why do philosophers consider their memory and perception as a valid source of knowledge but not their intuition? Aren't memory, perception and intuition (pre)processed/coloured by the same unconscious processes before we get aware of it? Kobe

What has happened in the last hundred years which convinces us that our 'scientific knowledge' is any more valid than previously?

Is scientific research a good use of government funding when hospitals, schools and social services are suffering from tight budgets??

What are some real-life examples using reason (deductive or inductive) in a sound and valid manner and coming up with a false statement of reality? In other words, I'm trying to prove that reason is not always a reliable way of knowing.

Can you think of a single justification for your existence that Harry Potter couldn't use? "I think, therefore I am" doesn't work, because Harry thinks, but doesn't exist.

Considering Descartes' malicious demon idea, is it possible that we could be manipulated in such a way so as all our beliefs are false? I'm thinking that we'd already need some true beliefs in order to have false ones. To be fooled into thinking that pig beards are shorter on Tuesdays I'd have to have true beliefs about pigs, beards, length, and Tuesdays for example. Can I infer then that the overwhelming majority of our beliefs must be true?