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By definition, the conclusion of any deductive argument follows directly from the premises. For example, consider the following famous syllogism:

Premise 1 - All men are mortal.

Premise 2 - Socrates is a man.

Conclusion - Therefore, Socrates is mortal.

Notice that the conclusion doesn't state anything new; it is just a restatement of information contained in the two premises. All of modern mathematics is based on this type of reasoning. Because of this, mathematics should be obvious since it just restates things that we already know.

Yet mathematics is not obvious, as many mathematical discoveries are surprising. How can a style of argument that just restates the premises be considered to increase our store of knowledge in any non-trivial way?

closed as off-topic by Mauro ALLEGRANZA, jeroenk, Keelan, iphigenie, Hunan Rostomyan May 8 '15 at 1:50

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  • "Questions that push a personal philosophy with no question beyond "am I right" or "what do you think" are off-topic here as this is not a blog. It's ok to express unique opinions, but you must have an actual, answerable question to go with them." – Mauro ALLEGRANZA, jeroenk, Keelan, iphigenie, Hunan Rostomyan
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    Is there any chance you could clarify what you think might be of philosophical interest/significance in this question? (What have you been reading or studying that has made this problem an important one in your study of philosophy?) – Joseph Weissman May 7 '15 at 14:06
  • People tend to find things interesting when they are filled with surprises. There are no surprises in mathematics. Then why do many people find mathematics interesting? – Craig Feinstein May 7 '15 at 14:10
  • Can you please clarify your question. I suspect your are asking something very interesting regarding a priori/a posteriori and synthetic/analytic truths. But your question isn't very clear. It might help to reword it. – Alexander S King May 7 '15 at 14:26
  • I'll think about it. You are correct that it is about a priori/a posteriori and synthetic/analytic truths. – Craig Feinstein May 7 '15 at 14:29
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    @CraigFeinstein your question was a good one, but I think the negative response was due to words like "boring" which sound judgmental and seem to call for an opinion. Avoiding words like that for future questions will increase the odds that your question will be positively received and won't get closed. – R. Barzell May 7 '15 at 16:35
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You are correct that the conclusion of any valid deductive argument contains at most a subset of the information contained in the premises.

However, until you work through the proof you don't know if the premises and conclusion actually hold that relationship to one another. What you gain via the deductive argument is new information about the equivalence of things not previously demonstrated to be equivalent. This can be very useful because it uncovers patterns and relationships that would otherwise be disguised by the superficial differences in the way the equivalent information is expressed.

  • I would qualify the first sentence, at least where mathematics and mathematical logic are concerned. Conclusion of a deductive argument depends not only on premises, but also on axioms and rules of inference. Even if axioms are lumped together with premises conclusions under intuitionist rules will be different from classical ones, and in first order logic will be different from in second order logic. – Conifold May 7 '15 at 20:55
  • Kantian thesis that deductive argument adds no new information only works if by "logic" one means Aristotle's syllogistics, and by "information" one means knowledge available to an omnipotent Deity. Modern definition is relativised, deductive argument is not much different from extracting the message from a cipher, it's "all there", but the message if far more informative than the cipher. – Conifold May 7 '15 at 21:03
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First, is deriving new facts from existing ones just restating the premises?

Second, why shouldn't knowledge be surprising or useful just because it starts with known givens? For instance, a sculptor uses tools to carve a statue from a block of marble following a specific process. Well, everything was "known" at the outset and the statue was certainly implicit in the marble, yet it's still an achievement. Math and logic follow the same analogy...

  • Axioms → the block of marble
  • Derivation Rules → hammer & chisel
  • Deduction → process of carving

Third, they give us new knowledge. Trying to strictly reason can reveal our own biases, mistakes and assumptions, and that is new knowledge.

Finally, the fact that math and logic have given us powerful tools to solve problems that we were otherwise unable to solve flies in the face of them being merely "restatements of knowledge".

In short, don't think of math or logic as restating knowledge; think of them as generating new knowledge using existing knowledge and strictly defined derivation rules.

This whole question of knowledge and what we already know reminds me of Plato -- namely Meno's Slave.

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The root issue that is occurring here relates to knowledge and implication.

Knowing both

(1) A 
(2) A -> B 

does not turn out to be identical to also knowing B. You're missing that in stating the "Notice that the conclusion doesn't state anything new" and "it just restates things that we already know."

At least for human beings our knowledge seems to work in such a way where we don't always draw the material implication from our set of assumptions. Instead, we have to engage in thought to get there.

In other words, my claim is that math and other disciplines do involve adding knowledge, because knowing what would imply that you should know something else is not identical to actually knowing that something else. (It would be for a perfect logic machine or for God, because the mode through which they know things is different from our process of thought).

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Firstly, you know it is contained in the two premises because of mathematics. Without mathematics, you can't know Socrates is mortal from the two premises, and that information will be new to you. (They are not really new, but you never know, and we are talking about how you are enjoying them.) You can argue that some mathematics results may be boring if spoken loudly. But depending on how you define "boring", the existence of mathematics at the first place is either very interesting, or neither in essence boring nor interesting because your definition depends on them and assigned them a base boringness value in itself. It also increases accuracy upon formulating the mathematics, which is a kind of information. And some knowledges in mathematics like "something is strictly impossible" is hardly being deducible from real life experience.

Secondly, after you had the informations, it can still cost much to find them. You have to look up the information in the infinitely many mathematic truths whenever you want to use them. The informations in mathematics isn't organized well to satisfy real world uses easily. Studying mathematics can put the hidden truths into obvious positions. And after you have done, the process will be irrelevant in further problems, or you can even make the solutions to some problems automated. At least they reduces a lot of pain, sometimes in creative ways.

Finally, mathematics give you freedom to work with things those are never happened. You can understand an object without risking breaking it. And you can derive good things from evil without really committing evil. They are simply not truths or tautologies, and may contain information, even though the conclusions (like what happens if they become true) may be boring in your definition.

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