Can a fixed procedure for reasoning settle a disagreement when the participants accept the same premises?

The ancient syllogism was incapable of mechanical performance because of its extreme incompleteness and crudeness, and it is only when we found our system upon the fundamental laws of thought themselves that we arrive at a system of deduction which can be embodied in a machine acting by simple and uniform movements.

It is probably this important fact along with other philosophical reasons that gives rise to the conviction (which every mathematician shares, but which no one has as yet supported by a proof) that every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts. Take any definite unsolved problem, such as the question as to the irrationality of the Euler-Mascheroni constant , or the existence of an infinite number of prime numbers of the form . However unapproachable these problems may seem to us and however helpless we stand before them, we have, nevertheless, the [Pg 9] firm conviction that their solution must follow by a finite number of purely logical processes. … This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus .

They are scarcely seen; they are felt rather than seen; there is the greatest difficulty in making them felt by those who do not of themselves perceive them. These principles are so fine and so numerous that a very delicate and very clear sense is needed to perceive them, and to judge rightly and justly when they are perceived, without for the most part being able to demonstrate them in order as in mathematics; because the principles are not known to us in the same way, and because it would be an endless matter to undertake it. We must see the matter at once, at one glance, and not by a process of reasoning, at least to a certain degree. And thus it is rare that mathematicians are intuitive, and that men of intuition are mathematicians, because mathematicians wish to treat matters of intuition mathematically, and make themselves ridiculous, wishing to begin with definitions and then with axioms, which is not the way to proceed in this kind of reasoning. Not that the mind does not do so, but it does it tacitly, naturally, and without technical rules; for the expression of it is beyond all men, and only a few can feel it.

"You might," the candid hero admitted; "though such obtuseness would certainly be phenomenal. Still, the event is possible . So I must ask you to grant one more Hypothetical." "Very good. I'm quite willing to grant it, as soon as you've written it down. We will call it ( D ) If A and B and C are true, then Z must be true. "I have !" Achilles joyfully exclaimed, as he ran the pencil into its sheath. "And at last we've got to the end of this ideal race-course! Now that you accept A and B and C and D , of course you accept Z ." "Do I?" said the Tortoise innocently. "Let's make that quite clear. I accept A and B and C and D . Suppose I still refused to accept Z ?" ​ "Then Logic would take you by the throat, and force you to do it!" Achilles triumphantly replied. "Logic would tell you 'You can't help yourself. Now that you've accepted A and B and C and D , you must accept Z !' So you've no choice, you see." "Whatever Logic is good enough to tell me is worth writing down ," said the Tortoise. "So enter it in your book, please. We will call it ( E ) If A and B and C and D are true, Z must be true. Until I've granted that , of course I needn't grant Z .

Should your rules be followed blindly? Yes, else only intuition could enable us to distinguish among them; but then they must be infallible; for only in an infallible authority can one have a blind confidence.... Here it is not at all the case; the logicians have applied their rules, and they have fallen into contradiction; and so true is this, that they are preparing to change these rules and to "sacrifice the notion of class."