The conjecture has been proved by induction. There is, as yet, no deductive argument that also proves it. Mathematics recognizes proof by induction, drkitten.drkitten said:
This is verging on the silly, but it's also typical of your rather short-sighted view of the practical aspects of mathematical thought.
Have you ever noticed that lots of even numbers (greater than four) appear to be the sum of two odd primes?
6 = 3 + 3
8 = 5 + 3
10 = 3 + 7 = 5 + 5
12 = 7 + 5
100 = 97 + 3 = 93 + 7 = 89 + 11 = ...
Induction suggests a hypothesis : Every even number (greater than 4) is the sum of two odd primes. (This, of course, is the famous Goldbach conjecture.)
Induction, as shown, will generate the hypothesis, but to prove the hypothesis will require deductive reasoning. If you figure out a suitable deductive argument "proving the hypothesis," let me know. I -- and most of the number theory community -- would be interested.
q.e.d.?
(Edited to fix rather embarassing typo.)
BillHoyt said:
The conjecture has been proved by induction. There is, as yet, no deductive argument that also proves it. Mathematics recognizes proof by induction, drkitten.
drkitten said:
The Goldbach conjecture has NOT been proven true by induction; it hasn't been proven true at all, and there's possibly a Fields medal in it for you if you can show it to be true. Experimental evidence has proven it to be true up to about 10^14, but of course, it's a long way from there to infinity.
More directly, "induction" and "proof by induction" are two different things. Proof by induction is actually an example of deductive reasoning, where you use the two premises that F(0) is true and that if F(i) is true, and so infer deductively that it must be true for all i (>= 0). Mathematics recognizes proof by induction only because it's deductively valid.
Which is the point. We have strong inductive evidence that the Goldbach conjecture is true, but no amount of inductive evidence will demonstrate its truth. To do that will require deductive reasoning. Induction will generate the conjecture, but few conjectures can be proven by mere enumeration of cases (which is all induction really is).
BillHoyt said:
No, that is MY point, drkitten. In Logic, deduction does not prove induction. Induction stands on its own. We are, then, left with the dilemma: induction creates new knowledge, but doesn't guarantee the knowledge, and deduction guarantees the knowledge, but can't create new knowledge.
Induction, going from the specific to the general, supplies a hypothesis, or a premise. That process of producing hypotheses is different than the deduction used to prove a hypothesis.
The process of producing the premise is different than the process that proves the premise.
Induction can supply you with a new theory. Deduction can assist you in analysing it. One does not become the other.
Keep in mind the objective is to get to new knowlege. Induction gets new knowledge, but can't guarantee it. Deduction guarantees knowledge, but can't produce new knowledge. Yeah, I'd call that a dilemma.drkitten said:
That's a dilemma?
"We're left with this dilemma: hammers pound nails, but can't turn screws, and screwdrivers turn screws, but can't pound nails." You use the tool to the purpose at hand.
Stop restricting us to mathematics. Stop restricting us to the Goldbach conjecture. The topic is logic's ability to aid us in making decisions. Please go back to the beginning of this thread to see that is so.Returning to IndigoRose's statement
... I think it's quite clear that she's absolutely correct. The Goldbach conjecture example demonstrates it. Induction will produce the hypothesis, but the deduction used to prove is [will have to be] different.
As she said elsewhere:
and
Perhaps you can explain your objections to those three statements of hers in light of the Goldbach conjecture example? [/B]
BillHoyt said:
Keep in mind the objective is to get to new knowlege. Induction gets new knowledge, but can't guarantee it. Deduction guarantees knowledge, but can't produce new knowledge. Yeah, I'd call that a dilemma.
Stop restricting us to mathematics. Stop restricting us to the Goldbach conjecture. The topic is logic's ability to aid us in making decisions....
And yes, my objections stand and are quite clear.
Induction, going from the specific to the general, supplies a hypothesis, or a premise. That process of producing hypotheses is different than the deduction used to prove a hypothesis.
Really? One wonders why science was invented.drkitten said:
So you use induction to get to new knowledge, and then you use deduction to confirm the truth of what you suspected from induction.
This is elementary, and hardly a dilemma.
So one can deduce that all organisms on earth use DNA or RNA? Please explain how that is so.Then perhaps you could actually restate your objections in a way that applies to a specific problem. I'm hardly "restricting" you to mathematics, but if your epistemological framework won't even let you make logical decisions in as ritualized and restricted a framework as mathematics, then there's something deeply wrong with your framework.
If you prefer your own (nonmathematical) example, you drew the conjecture that "all day lillies are orange" from your observations of fields of day lillies. As you yourself acknowledge, that statement has not been "proven" true. If you want to prove it true, however, it's (relatively) easy to set up a deductive argument that produces that proposition as a conclusion. For example,
All plants with this particular gene produce this protein to color their flowers.
This protein colors flowers orange.
All day lillies have this particular gene.
Therefore, all day lillies are orange.
Without the impetus provided by the induction, you likely would never have thought to look at the genetics of flower coloration. Thus, induction provided you with the conjecture, and deduction (albeit from a different set of premises) guarantees the truth of that conclusion. Induction and deduction together provide you with a method of getting new knowledge and making decisions.
Or, as IndigoRose put it (and you have not yet substantially objected to) :
BillHoyt said:
Really? One wonders why science was invented.
But you just wrote: "So you use induction to get to new knowledge, and then you use deduction to confirm the truth of what you suspected from induction." There seems to be no need for science's methods here. Induction and deduction would seem to be enough.drkitten said:
Because previous methods of doing "natural philosophy" had relied almost solely on deductive reasoning. "Science" is simply the most reliable way we've found for merging inductive and deductive reasoning.
For a longer answer, I refer you to Sir Karl Popper.
BillHoyt said:
But you just wrote: "So you use induction to get to new knowledge, and then you use deduction to confirm the truth of what you suspected from induction." There seems to be no need for science's methods here. Induction and deduction would seem to be enough.
I again pose to you the question: why science? You claim deduction is used to prove induction. This is logically incorrect. You claim natural philosophy relied solely on deduction. This is not so. I have asked you to explain why science was necessary and you claim it merges induction and deduction But this flies in the face of your claim that deduction proves induction. I await a direct answer.drkitten said:
I again refer you to Sir Karl Popper.
drkitten said:
Have you ever noticed that lots of even numbers (greater than four) appear to be the sum of two odd primes?
6 = 3 + 3
8 = 5 + 3
10 = 3 + 7 = 5 + 5
12 = 7 + 5
100 = 97 + 3 = 93 + 7 = 89 + 11 = ...
Induction suggests a hypothesis : Every even number (greater than 4) is the sum of two odd primes. (This, of course, is the famous Goldbach conjecture.)
Induction, as shown, will generate the hypothesis, but to prove the hypothesis will require deductive reasoning.
Induction, going from the specific to the general, supplies a hypothesis, or a premise....
BillHoyt said:
I again pose to you the question: why science? You claim deduction is used to prove induction.
Sorry to hear you are unable or unwilling to answer.drkitten said:
Objected to as already asked and answered.
Again, I refer you to Sir Karl.
BillHoyt said:
Sorry to hear you are unable or unwilling to answer.
I don't think you should presume I haven't already. A reference to an author hardly answers a specific question. That author may have in his many works, provided an answer, but it is incumbent upon you to specifically quote that author or to paraphrase his answer.drkitten said:
I'm unwilling to repeat the answer I've already given. Please try again. After you've read Sir Karl Popper, who has addressed these issues in depth.
BillHoyt said:
I don't think you should presume I haven't already. A reference to an author hardly answers a specific question. That author may have in his many works, provided an answer, but it is incumbent upon you to specifically quote that author or to paraphrase his answer.
Anything less is not only a dodge, but a personal insult.
Here is what you said:drkitten said:
I've already paraphrased his answer above, and you misread it. Therefore, I refer you to the master : Sir Karl Popper.
Not true in the least. Newton's Principia, for example, offered this rule: "In experimental philosophy we are to look upon propositions collected by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phænomena occur, by which they may either be made more accurate, or liable to exceptions"Because previous methods of doing "natural philosophy" had relied almost solely on deductive reasoning.
This doesn't address the question of your claim about induction and deduction's sufficiency."Science" is simply the most reliable way we've found for merging inductive and deductive reasoning.
BillHoyt said:
Not true in the least. Newton's Principia, for example, offered this rule: "In experimental philosophy we are to look upon propositions collected by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phænomena occur, by which they may either be made more accurate, or liable to exceptions"
Clearly, Newton here is recognizing general induction as the basis for "experimental philosophy's" (another name for natural philosophy) propositions.
I agree that this was part of the breakaway of Natural Philosophy from philosophy. This began a great war that ultimately resulted in Science divorcing itself from Philosophy in many respects. Newton called it induction to be sure, as have many others since, but it is not quite so.drkitten said:Right. That's because Newton is one of the early founders (and inventors) of the scientific method, along with his close contemporaries such as Gallileo, Copernicus, and the other leading lights of that era. If you compare that method of inductive inquiry with the earlier work, for example, of the Scholastics or of the Greek philosophers, there was a general opinion that (deductive) reason alone would explain the natural world, without recourse to experimentation. Newton, in this excerpt, is basically defending the as-yet unnamed concept of "the scientific method" against the previous philosophical concepts.
True enough. There was more going on there, though. At the time, the nascent scientists were fighting a battle on at least two fronts. One was with Religion, whose tenets were already being challenged, even though Science and scientists both strove to serve God. They saw themselves as trying to reveal His creation. But they served another master as well: probity. They trusted their processes of tearing apart the watches to see what made them tick. This didn't sit well with the religious establishment.You can see further evidence for this shift in the mere fact that he felt compelled to write these paragraphs. In the same way that historians infer the existence of a practice from written documents decrying or prohibiting the practice, Newton's spirited defense of the specialized field he terms "experimental philosophy" illustrates the dominance at the time of what might be called "unexperimental philosophy," the rationalist/Scholastic doctrine of the sufficiency of reason, the doctrine of belief in "contrary hypothesis that may be imagined," but not supported via empirical evidence.
But this cannot and does not work, as will become clearer later. Let us say, for example, that we actually did induce the hypothesis that all swans are white. Then we devise a test wherein we raid a local pond, and grab a few swans. We test swan 1. White. Swan 2. White. Swan 3. White. There is no serialization here, unlike mathematical induction, whereby we can declare all swans to be white from this. Each deduction has simply said this is another instance of a white swan. When we finally encounter Swan 5,431 and find it is black, our only recourse is to modify the proposition to some swans are white, or, more boldy, most swans are white.The essentials of the scientific method are here in Newton's exposition : assume the provisional truth of propositions inferred inductively, but confirm their truth via other methods. Newton does not state that deductive logic is to be one of those other methods, but it's implicit in the rest of his writings, including the Principia itself.
Now we're at Popper, who first stated this proposition. Now he, too, called it induction, but we'll get to that error later. What you miss here in this description is that the deduction is NOT being used on the real hypothesis, but on a straw man hypothesis. The straw man was designed to be refuted, in order to drop one more possibility from the infinite set of possibilities. We never actually apply deduction to the hypothesis we are after. This is not a case of a deduction proving an induction.Since Newton, of course, "the scientific method" has not only become the dominant paradigm of reasoning, but has also been refined and improved. One of the weaknesses of the scientific method, for example, is that general induction, combined with the providional assumption of truth for propositions so obtained, is too weak a gatekeeper. A big role of deduction is thus to refute these presumed truths via what amounts to modus tollens.
Actually, no. Peer review simply weeds out obvious errors in method or obvious disconnects between hypothesis tested and conclusions claimed.It would be more efficient if we had a way to reduce the number of incorrect hypotheses accepted. Thus, there are a number of technical improvements that have happened since then. The peer-review system allows scientists to confirm that the inductive reasoning used by the original science is in fact valid.
No, these methods are only used in the softer sciences, where self-reporting, interpretation and possible unconscious bias are large effects. They are not part of the overall scientific framework.The idea of a "control group" provides some degree of protection against false induction. A further protection was provided by the idea of blinding, and then double-blinding, experiments to avoid known effects that would be likely to result in the false acceptance of a bad hypothesis.
I think you forget that the number of possible hypotheses is infinite. This is true for every proposition you wish to test.Of course, deduction could eliminate these hypotheses (when the presumed-true theory was shown incompatible with other data), but why not just elliminate the extra step and apply a higher standard to inductive reasoning?
What we are talking about here, as I've stated previously, is not deduction about the target proposition, but deduction about a tangential straw man. And what we are talking about here is not induction in the first place, but something else. Review the science literature in any specialized area. You will not find a succession of papers that each simply report an instance of X, followed by a paper testing the now established induction of X as a general proposition. X is actually created by retroduction, also known as abduction, a third leg of the logic ladders."Science," thus, is just the current best way we've found to merge inductive and deductive reasoning. The basic framework was laid by Popper; theories are generated by whatever means, usually inductively, through the inspection of data. A theory is really only "scientific" if it makes a testable prediction, that is, if deductive reasoning applied to the theory results in a statement that can be shown to be true or false. A subsequent finding that the prediction is false implies (deductively, via modus tollens) that the original theory itself is false.
Add in a few PDF files for grant proposals, and you've re-invented the National Science Foundation.
What aspect of science do you feel is NOT captured by applications of inductive and deductive reasoning?