PJPSYCO

Then you tell him, "Why do you keep arguing? If nothing exists then there is nothing to argue about."

Theli

But is R + 1 really a number?
I mean if R is a variable then R + 1 can't be regarded as a number as it has no set value.

SlateGreySky

If your friend is asking for certainty beyond all conceivable doubt, tell him you have none to give. Then ask him what certainty he can offer over and above the definitions of terms.

If absolute certainty is your friend's criterion for knowledge, he'll find that he knows very little indeed. The fact is, "knowing" doesn't work that way: we don't ask for indubitable certainty when we ask for knowledge; it would be futile.

I've found that a good first step in discussions like these is to ask, "what would you accept as proof" or "by what criteria are you judging 'knowledge'?" By asking that question, you forgo a good deal of argumentation in which both parties miss each other because each party is playing by a different set of rules.

Best of luck!

Goober

No, there is no such thing as the highest rational number. Proof:

Assume the highest rational number exists, and call it n.

But 1>0, therefore n+1>n (adding n to both sides, as n>0 or else it would clearly not be the largest rational number).

Also, a number is rational if it is expressible as a/b where a and b are integers. If n=a/b where a,b are integers, n+1=a/b +1=(a+b)/b. The sum of two integers is an integer therefore a+b is an integer, so n+1 is rational and larger than n.

But this is a contradiction, because n is the largest rational number, so our initial assumption is false and there is no largest rational number.

Of course this is only true if the axioms of mathematics are true. However, the statement 'IF the axioms of maths are true, then there is no largest rational number' is always true, so yes there are some things that can be known as certainty.