RoR: Forum

I addressed induction here.

Post 1

Sunday, January 20, 2013 - 9:19am

Joe,

I like how clear you have made things. In conclusion, you cite 2 kinds of generalizations, which could be called:

1) simple
2) formal

A simple generalization is what you usually see in modern science, a progressive enumeration of causes, widdling away at some of them while eventually accentuating others (as more information reveals that these others were more primary or fundamental). In contrast, a formal generalization would be a completely valid (read: a "sound") induction -- as I illustrated here, eventually settling down to calling it something like: completely-valid generalization, arrived at via adequate mechanical limitation.

Do you think this is possible?

In the case of swans, it'd include being able to say what it is that swans will always be, or be like -- and also what it is that they never will be, or be like (e.g., black, white, avian, mammalian, herbivorous, carnivorous, terrestrial, aquatic, endothermic, exothermic, etc.).

Ed

(Edited by Ed Thompson on 1/20, 9:21am)

Post 2

Sunday, January 20, 2013 - 10:36am

Yes, Ed, Rowlands did a nice job of amplifying Harriman's argument in The Logical Leap.
Statistical inferences - enumeration - can be intuitively helpful but are not the way to establish truth. For something to be true, it must be empirically verifiable and with a logically consistent explanation. "All swans are white." has only part of the first and none of the second. Quantum mechanics, on the other hand, has both.

Merlin's essay was more formal. As far as I can tell the two do not contradict each other on any point.

I did like Merlin's closing with a quote from Francis Bacon. This is an old and known problem.

Francis Bacon wrote: "The induction which proceeds by simple enumeration is puerile, leads to uncertain conclusions, and is exposed to danger from one contradictory instance, deciding generally from too small a number of facts, and those only the most obvious" (Novum Organum, I 105).

(Edited by Michael E. Marotta on 1/20, 10:40am)

Post 3

Tuesday, January 22, 2013 - 10:58pm

Thanks for the comments. This article wandered a little, but the topic is difficult to pin down.

Ed, if I understand your "formal" generalization, it sounds like the generalized is based on deductive reasoning, not inductive. The example you gave (if I understood) in your article was about chemical reactions. The generalization that one element won't chemically bond with another is deduced from the theory of valence electrons.

That's fine. Valid generalization can be deduced from wider generalizations. But that approach is not a substitute for induction. It can't be. Deduction requires generalizations to begin with.

Michael, you mention Harriman's book. While I certain find some points of agreement with that book, I think the overall theory is flawed, as I mentioned in another thread some time ago.

Induction is messy. It is entirely possible to form an incorrect generalization not because of bad method, but because of limited information. And those faulty generalizations are actually wrong, and over-contextualizing them doesn't help. The whole point of generalization is to form conclusions that go beyond the scope in which the generalizations are formed.

It's also worth noting that while an induction should be logically consistent with the rest of your knowledge, that doesn't mean it is logically necessitated. Again, that would be a deduction.

Post 4

Wednesday, January 23, 2013 - 10:40am

Joseph, I found your original Objectivism 101 article. (I also found your essay on Axioms.) Please post a link or a citation to the essay you would like me to read. Essentially, I did not see that much difference here between your explanation and Harriman's. Thanks.

Post 5

Wednesday, January 23, 2013 - 7:19pm

Joe,

Let's create a couple scenarios wherein induction is used, making the matter concrete. Upon doing that, there will be clear times where one could say:

Do you see what 'Phil the Philosopher' (or Sammy the Scientist) is doing right here? He is using deduction right here. This is because ...

Do you see what 'Phil the Philosopher' is doing right here? He is using induction right here. This is because ...

You said that induction is messy, but I do not think it is inherently messy. Instead, if it is still true that induction appears messy, then I think that people have accidentally had poor explanations of induction in the past -- making it merely appear to be something that is inherently messy. If I am correct, then concrete examples should be able to elucidate that (and if I am wrong, the examples will show that, too). We can create our own situations out of thin air, or we could use historic examples. Below, I provide 10 historic examples from the Wiki entry for Timeline of scientific experiments.

Ed

10 historic, concrete examples of the human use of induction

240 BC � Eratosthenes measures the Earth's circumference and diameter
1609 � Galileo Galilei observes moons of Jupiter in support of the heliocentric model
1638 � Galileo Galilei uses rolling balls to disprove the Aristotelian theory of motion
1798 � Henry Cavendish: Torsion bar experiment to measure the gravitational constant
1801 � Thomas Young: double-slit experiment showing wave-particle duality
1843 � James Prescott Joule measures the equivalence between mechanical work and heat, resulting in the law of conservation of energy
1845 � Christian Doppler demonstrates the Doppler shift
1861 � Louis Pasteur disproves the theory of spontaneous generation
1863 � Gregor Mendel's pea plant experiments (Mendel's laws of inheritance)
1952 � Alfred Hershey & Martha Chase: Hershey-Chase experiment proves that DNA is the hereditary material

(Edited by Ed Thompson on 1/23, 7:22pm)

Post 6

Thursday, January 24, 2013 - 1:19am

Michael, I made several posts on the topic under the book review for The Logical Leap. Here.

To summarize, I'm suspicious of how he focuses exclusively on cause and effect relationships. I also disagree with the "first level generalization" approach. The approach seems borrowed from concept formation. Start with the given, which is perception. Abstract from there. But I don't it's a valid approach. I think his reasoning about first level generalizations is flawed. And his entirely theory rests on those being completely validated so he can abstract from there.

Ed, I think induction is messy. Consider deduction for a moment. You start with the premises and combine them, according to the rules of logic, to form some new conclusion. It's all very self-contained at isolated from the rest of your knowledge.

Induction, on the other hand, rests on the breadth of your knowledge. The more you know, the more justified you can be in making an inductive leap. But that means it's also possible that there is knowledge you are lacking that would invalidate it. It doesn't have the simplicity and isolation of deduction. It takes significant integration and knowledge.

One of the problems with many of the examples you offered, and in science history in general, is that there are often a single significant experiment that seemingly answered the question. But that's not representative of induction. Some (maybe all)of those are deduction. You take a single data point, combine it with a wide principle, and deduce that one theory is right and another is wrong.

Trying to deal with the problem of induction by referring to situations where a single data point is enough to reach a definitive conclusion is questionable. Even if it was induction, it seems aimed at trying to make induction more like deduction. It tries to get rid of all of the uncertainty and difficulty of justifying the inductive leap through a massive amount of information and integration.

The real question with induction is how you get to those wide-reaching principles. How do you go from a small set of data points to a wider generalization? And if you have to refer to an even wider generalization to get there, it's deduction not induction.

As I said, I think it is messy and it's a fallible process. And that's not just because of the possibility of misusing rules of reasoning. It is because induction is a process that rests on a large body of knowledge, and if you are missing crucial information, no amount of diligence can get you to the right conclusion.

Post 7

Thursday, January 24, 2013 - 5:02am

Joseph and Ed: Mendel's observations qualify as induction because he carried out many over time to establish a body of information. In fact, it so happens that the "tall pea / short pea" dichotomy is not as neat as portrayed. As for the others, they are examples of deduction.

Eratosthenes could not have measured the circumference without geometry, a purely deductive science.

Among the many brilliant points in Newton's work, perhaps the "central" was his proof that central force motion requires and is required by Kepler's laws. We teach that today as conservation of energy. Central force motion is conservative. Here on Earth, as a body falls, potential energy changes to kinetic energy - and with a pendulum they exchange. (It also applies to thermodynamics, as noted, but Joule knew by deductive theory what he was looking for.)

Just a note: Thomas Young's experiment did not demonstrate "wave-particle duality." It only established the wave nature of light. However, the particle nature of "light" also was known from experiments and was further validated for the next 100 years first as atoms, then electrons, neutrons, X-rays, etc., were discovered.

Another note: Pasteur did not disprove spontaneous generation. Absence of evidence is not evidence of absence. But I agree that we came to the germ theory by the inductive accumulation of evidence. The microscope was known from the 17th century - lenses go back even farther: we think that the ancient Greeks knew magnifiers. The argument is whether the germs show up as a consequence of the illness caused by a "miasma." Malaria means "bad air." We all know that. (Walter Reed's work came after Pasteur's of course.) ( My article, "From Texas to the Moon with John Leonard Riddell")

The real "problem of induction" is a problem that some Objectivists have with the non-absolute and the uncertain. It is why some argue against quantum mechanics and relativity.

Allow me just a word about the Identity side of this. We are taught that abstract mathematics has no real application. In fact, I believe that such work has only not found an application; but any mathematics, if deductively valid, does describe physical, empirical reality. The ancient Greeks denied "irrational" numbers and negative numbers. For centuries the square root of minus one was "imaginary." Mathematics is like one of those big red rolling Craftsman tool boxes: even if you do not know what a tool is for, it still works on some aspect of reality. I was pleasantly surprised when I replaced a little belt in the head of a Hoover vacuum cleaner to see that it went on the drive and take-up to make a Moebius strip.

Joseph: I will go back and read those. Thanks.

(Edited by Michael E. Marotta on 1/24, 5:12am)

Post 8

Friday, January 25, 2013 - 6:44pm

Joe,

Point taken. As a side-note, I think it better to refer to induction as a species of inference, like deduction is. There is inductive inference and there is deductive inference, but whenever you talk about either -- you are talking about an inference. Sense-perception is different in that it is non-inferential, so there are 3 things in play:

1) perceptual pick-up of variation in your ambient stimulus array (e.g., seeing a piece of wood floating in water)
2) deductive inferences made from perception or from other inferences or from perception combined with other inferences
3) inductive inferences made from perception and always combined with other inferences (though possibly only from perception combined with strictly deductive inferences)

It's a simple point but I think it might help, and I don't recall in my readings any philosopher who worked to make it really explicit (though Robert Audi did a pretty good job at this in his book: Epistemology).

Ed

Post 9

Friday, January 25, 2013 - 7:25pm

[A deleted double-post.]
(Edited by Ed Thompson on 1/25, 7:29pm)

Post 10

Friday, January 25, 2013 - 7:26pm

Mike,

Eratosthenes could not have measured the circumference without geometry, a purely deductive science.

Okay, but just because part of an experiment may have involved a deduction does not make a conclusion -- the drawing of an inference -- from the experiment deductive. Deduction is not like some kind of a plague which transforms everything it touches into being a deduction, too. Instead, it is a tool. A pretty good heuristic for deduction is that it aims at the particular, rather than aiming at the general. A pretty good heuristic for induction is that it starts with a particular, rather than starting with a general. Here are examples:

Deductive

Coins have 2 sides.
[This] is a coin.
---------------
Therefore, [this] has 2 sides.

The above is a deductive inference which starts with the general, but aims at the particular.

[This] is a coin.
[This coin] has a "heads" side and a "tails" side.
------
Therefore, the "heads" side of this coin is one side of a coin.

The above is a deductive inference which starts with the particular, and aims at the particular.

Inductive

A shadow lengthens as the sun lowers on the horizon.
One of the possible ways for this to be the case is if light rays from the sun are blocked by the material of the object casting the shadow.
------------------------
Therefore, a possible explanation of lengthened shadows corresponding to lowered sun angles, is that it is a general matter of the course of events that objects block at least some of the rays of light. And, if that were the case, then shadows tomorrow will respond the same as shadows today, and shadows of different objects will respond at least roughly the same as well.

The above is an inductive inference which starts with the particular, but aims at the general.

This statue casts a shadow that is 3-feet long at midday on November the 23rd.
One of the possible ways for this to be the case is if the earth is in a year-long orbit around the sun and, therefore, the sun ends up in precisely the same position in the sky at precisely the same time on precisely the same day each year.
---------------------------
Therefore, a possible explanation of why this statue's shadow is exactly 3-feet long at this precise time on this precise day of the year, is because the earth is in a year-long orbit around the sun (which puts the sun in exactly the same sky position at that time of day, once every 365 days).

The above is an inductive inference which starts with the particular, and aims at the particular.

Ed

Post 11

Saturday, January 26, 2013 - 5:23am

Mike,

Just a note: Thomas Young's experiment did not demonstrate "wave-particle duality." It only established the wave nature of light

Good point.

Another note: Pasteur did not disprove spontaneous generation. Absence of evidence is not evidence of absence. But I agree that we came to the germ theory by the inductive accumulation of evidence.

Alright, alright. But it's not like we're stuck in a conundrum or something. Spontaneous generation involves something coming from nothing, and we do not need to look out into the world in an attempt to try to disprove such a thing:

Son: Dad, I'm worried I might get sick.

Dad: How?

Son: By the spontaneous generation of a causal factor for a disease, where none had existed before.

Dad: By something coming from nothing?

Son: Yes.

Dad: That's baseless. Stop worrying, Son. Instead, spend your time and energy worrying about things which -- by either evidence or reasoning --- are known or suspected to be true.

Son: Okay, I'll start doing that. I'll start limiting my focus to things which can rationally be known or suspected to be true, and I will dismiss things which cannot be rationally known or suspected to be true.

:-)

Ed

Post 12

Saturday, January 26, 2013 - 5:35am

Ed, Eratosthenes did not prove that the world is round by measuring the circumference. He knew that the world is round and measured its circumference. "Everyone" knew that the world is round. Aristotle summarized the evidence, but the evidence had been offered even earlier, of course.
Moreover, Eratosthenes did not make repeated measurements all over the world to establish inductively that the Earth must be spherical.

(Edited by Michael E. Marotta on 1/26, 5:50am)

Post 13

Saturday, January 26, 2013 - 7:02pm

Mike,

Eratosthenes did not prove that the world is round by measuring the circumference. He knew that the world is round and measured its circumference.

But where are you going with this line of reasoning?

Moreover, Eratosthenes did not make repeated measurements all over the world to establish inductively that the Earth must be spherical.

Well, from the looks of it, we probably disagree regarding the requirements of human induction. For you, it seems, you have to be making repeated measurements over a vast amount of time (as you said for Mendel) or space (as you say for Eratosthenes). This puts you on the spot somewhat: How much time or space is required in order to pass into the arena of induction? Would 3 years of repeated measurements or 3 square miles of them suffice? How about 6 of each?

If you want to set a bar like that for induction, then where do you draw the line? For me, you can perform an inductive inference on the very first go-around -- if you are savvy enough to pull it off.

Ed

Post 14

Saturday, January 26, 2013 - 10:52pm

http://www.regentsprep.org/Regents/math/geometry/GP4/PracCongTri.htm

Ed, Eratosthenes (200 BCE) knew the world was round because of a deduction, similar to the geometric proof below. It is more than the simple three-step syllogism which must be "true" no matter how silly: All grapes are clocks. All elephant are grapes. Therefore all elephants are clocks. The proof below is dedctive, but depends on the identification and integration of the correct knowledge. Several proofs of this same Given/Prove may be possible. The Pythagorean Theorem is a world favorite and many proofs exist for it. Each is a deduction, based on identification and integration. That is how Eratosthenes knew that the world is round: many arguments and reasons had been offered for its shape. Pythagoras (550 BCE) might have been the first (recorded) to say that Earth is a sphere.

I understand the problem of induction and the problem of enumeration. The fact remains that one instance might be enough and a million might not be. I went back and read all of Joseph Rowlands's posts in the thread on Logical Leap based on Merlin Jetton's review. I think that Joseph is not correct in all of his statements, but he does raise some interesting problems in how we know what we know.

What is most interesting to me in that is that his argument is itself an enumerative induction, citing example after example.

ET: "For me, you can perform an inductive inference on the very first go-around -- if you are savvy enough to pull it off."

Right, but how do you know? The examples you cited were not examples. Eratosthenes, Pasteur, and the others did not inductively prove what you claim they did. In many cases, true induction (enumeration) was called upon. In others, the claim was arrived at deductively, like the theorem below. Some of them were "logical leaps" also. Galileo and the Moons of Jupiter was one such because it answered the objection that the Earth would leave the Moon behind if the Earth orbited the Sun. However, he did not invent the Moons of Jupiter or go looking for them explicitly. He just found them and realized what they implied about the heliocentric theory. Still, it was, indeed one example of a logical leap. But as for the others... well... Just to take yet another, what did Henry Cavendish prove? Everyone knew that some constant was needed. And everyone knew what the units were. It was a nice bit of work, for sure, but it did not change any paradigm. Speaking of paradigns, have you read Thomas Kuhn?

... and by the way, Eratosthenes drew his map on a flat sheet of papyrus. He did not make a globe. So, he knew that his experiential world was but a small fraction of a very large body.

Question #6

Statements		Reasons
1.	;	1. Given
2.		2. Reflexive property - a quantity is congruent to itself.
3.		3. (ASA) If two angles and the included side of one triangle are congruent to the corresponding parts of a second triangle, the triangles are congruent.
4.		4. (CPCTC) Corresponding parts of congruent triangles are congruent.
5.	AB = BE ; DB = BC	5. Congruent segments are segments of equal length.
6.	AB = AD + DB EB = EC + CB	6. Betweeness of Points: The whole is equal to the sum of its parts.
7.	AD + DB = EC + CB	7. Substitution.
8.	AD = EC	8. Subtraction
9.		9. Congruent segments are segments of equal length.

Post 15

Sunday, January 27, 2013 - 12:01am

Michael, I agree with many of your points. The Mendel observation is a good case of induction, and highlights the potential to draw incorrect conclusions through limited data.

Ed, my earlier point was to be wary of redefining induction to avoid the problem of induction. How do you generalize from observations of singular events? If you generalize by combining concretes with wide-reaching generalizations, you are using deduction, not induction. Relying on generalizations in order to explain the creation of generalizations is circular. How do you get to the starting point?

On the example of Eratothenes, I agree with Michael that it is an example of deduction. There is no generalization of concretes. There is a deduction. Given a geometric relationship and given some number of data points, the you can deduce other data points. You may even be able to confirm those other data points. But it is an application of deduction.

I think your post 13 shows some confusion. Michael's point about the examples is not dismissing them because they don't have a sufficient number of data points. The issue is that the data points are used in a deduction instead of an induction. It is the method that is in question, not the number of data points.

This comes down to the question of certainty. If you think a single data point is enough to give you certainty, it seems you must be relying on a wider generalization that is deductively applied to this one case. Trying to call that induction avoids the real issues of induction.

Even if we consider a controlled experiment in a scientific laboratory, induction is never so simple. You can control for all known factors, but any conclusion you make is necessarily based on several assumptions. You have to assume that you didn't miss any factors. You have to assume that you successfully controlled the factors. And you have to assume your method of observing/measuring the result is accurate enough. Those are just three places the controlled experiment can go wrong.

If someone challenged your conclusion, you would have to explain why you believe the experiment was controlled well and why you think you covered all of the factors. But they might challenge other premises. To defend your conclusion, you could keep drawing upon a body of evidence to support your conclusions. But when that evidence is questioned, you might have to go even further.

The result is that induction has a significant foundation of knowledge that it draws upon. And all of that might be necessary to achieve the correct conclusion. The data points don't speak for themselves. They are understood in a context of substantial knowledge. And faults could be identified at any point in that foundation. New evidence could be discovered that changes any number of assumptions.

That's why induction is messy, and why too many philosophers (and Objectivists) try to downplay it or interpret it as a form of deduction. They crave certainty. But the nature of induction is that mistakes are possible. Trying to find a path to knowledge where mistakes aren't possible requires disconnecting ideas from reality.

Post 16

Sunday, January 27, 2013 - 4:05am

Okay, guys, but what about the inference that the cause of longer shadows on the ground is a lower sun in the sky, because the material of the objects casting the shadows is blocking light rays from the sun -- and such light rays emitted from lower angles will work to create longer shadows?

Is it really induction (as I say it is)?

Is it really something which can be known with at least contextual certainty?

Ed

(Edited by Ed Thompson on 1/27, 4:06am)

Post 17

Sunday, January 27, 2013 - 4:41am

Okay, guys, but what about the inference that the cause of longer shadows on the ground is a lower sun in the sky, because the material of the objects casting the shadows is blocking light rays from the sun -- and such light rays emitted from lower angles will work to create longer shadows?

Is it really induction (as I say it is)?

It looks like deduction to me. The two instances can be depicted as two right triangles, e.g. 3x4x5 and 5x4x6.403..., the middle 4 being the height of the object casting the shadow and the first number being the length of the shadow for both.

Post 18

Sunday, January 27, 2013 - 7:15am

JR: "That's why induction is messy, and why too many philosophers (and Objectivists) try to downplay it or interpret it as a form of deduction. They crave certainty. But the nature of induction is that mistakes are possible. Trying to find a path to knowledge where mistakes aren't possible requires disconnecting ideas from reality."

I mentioned that briefly above, that some (many?) Objectivists crave certainty, so they argue against Einstein's relativity, and Heisenberg's uncertainty; and physics aside, really do not feel comfortable not having an answer about something (or everything). But in truth, we learn more and new all the time. I pointed out elsewhere that when the Black Swan was discovered in Australia, it was correctly placed taxonomically with white swans. It was not the negation of knowledge or the disproof of induction or anything other than a new fact integrated into an existing understanding.

ET: Okay, guys, but what about the inference that the cause of longer shadows on the ground is a lower sun in the sky, because the material of the objects casting the shadows is blocking light rays from the sun -- and such light rays emitted from lower angles will work to create longer shadows?

MJ: Is it really induction (as I say it is)?It looks like deduction to me. The two instances can be depicted as two right triangles, e.g. 3x4x5 and 5x4x6.403..., the middle 4 being the height of the object casting the shadow and the first number being the length of the shadow for both.

Well, that... and the fact that the sun would make the same shadows if it circled the Earth or rose anew in the chariot of Apollo each morning and was put to sleep each night. In fact, the sun does "orbit" the Earth, if you want to think about it that way because no absolute observable frame of reference exists.

Oh... and Archimedes attempted to measure the parallax caused by the Earth orbiting the sun, as suggested by Aristarchos. However, he could not. He concluded that either the Sun orbits the Earth (as commonly believed) or else the universe is inconceivably large and the stars are very far away. So, even though scientific experiment suggested that the Sun orbits the Earth, Eratosthenes could calculate the circumference of the Earth because - as everyone pretty much agreed - the Earth is a sphere. (However, if the Earth were a "drum" like a coin as suggested by Democritos and others, it would still have a circumference, of course, and really, could be convex in surface, giving an illusion of sphericity to the Man from Cyrene.

(Edited by Michael E. Marotta on 1/27, 7:21am)

Post 19

Sunday, January 27, 2013 - 2:03pm