There are two particularly hard parts of explaining why induction is false. First, there are many refutations. Where do you start? Second, most refutations are targeted at professional philosophers. What most people mean by "induction" varies a great deal.
Most professional philosophers are strongly attached to the concept of induction and know what it is. Most people are strongly attached to the word "induction" and will redefine it in response to criticism.
In *The World of Parmenides*, Popper gives a short refutation of induction. It's updated from an article in Nature. It involves what most people would consider a bunch of tricky math. To seriously defend induction, doesn't one need to understand arguments like this and address them?
Some professional philosophers do read and respond to this kind of thing. You can argue with them. You can point out a mistake in their response. But what do you do with people who aren't familiar with the material and think it's above their head?
If you aren't familiar with this argument against induction, how do you know induction is any good? If you don't have a first hand understanding of both the argument and a mistake in it, then why take sides in favor of induction?
Actually, inductivists have more responses open to them than pointing out a mistake in the argument or rejecting induction (or evading, or pleading ignorance). Do you know what the other important option is? Or will you hear it for the first time from me in the next paragraph, and then adopt it as your position? I don't recommend getting your position on induction from someone who thinks induction is a mistake – all the defenses I bring up are things I already know about and I *still* consider induction to be mistaken.
Another option is to correctly point out that Popper's refutation only applies to some meanings of "induction", not all. It's possible to have a position on induction which is only refuted by other arguments, not by this particular one. I won't help you too much though. What do you have to mean by "induction" to not be refuted by this particular argument? What can't you mean? You figure it out.
Popper argues against induction in books like LScD, C&R, OK, RASc. Deutsch does in FoR and BoI. Should I repeat points which are already published? What for? If some inductivist doesn't care to read the literature, will my essay do any good? Why would it?
I recently spoke with some Objectivists who said they weren't in favor of enumerative induction. They were in favor of the other kind. What other kind? How does it work? Where are the details? They wouldn't say. How do you argue with that? Someone told me that OPAR solves the problem of induction. OPAR, like ITOE, actually barely mentions induction. Some other Objectivists were Bayesians. Never mind that Bayesian epistemology contradicts Objectivist epistemology. In any case, dealing with Bayesians is *different*.
One strategy is to elicit from people *their* ideas about induction, then address those. That poses several problems. For one thing, it means you have to write a personalized response to each person, not a single essay. (But we already have general purpose answers by Popper and Deutsch published, anyway.) Another problem is that most people's ideas about induction are vague. And they only successfully communicate a fraction of their ideas about it.
How do you argue with people who have only a vague notion of what "induction" is, but who are strongly attached to defending "induction"? They shouldn't be advocating induction at all without a better idea of what it means, let alone strongly.
There are many other difficulties as well. For example, no one has ever written a set of precise instructions for how to do induction. They will tell me that I do it every day, but they never give me any instructions so how am I supposed to do it even once? Well I do it without knowing it, they say. Well how do they know that? To decide I did induction, you'd have to first say what induction is (and how it works, and what actions do and don't constitute doing induction) and then compare what I did against induction. But they make no such comparison – or won't share it.
Often one runs into the idea that if you get some general theories, then you did induction. Period, the end. Induction means ANY method of getting general theories whatsoever. This vacuous definition helps explain why some people are so attached to "induction". But it is not the actual meaning of "induction" in philosophy which people have debated. Of course there is SOME way to get general theories – we know that because we have them – the issue is how do you do it? Induction is an attempt to give an answer to that, not a term to be attached to any answer to it.
And yet I will try. Again. But I would like suggestions about methods.
Induction says that we learn FROM observation data. Or at least from actively interpreted ideas about observation data. The induced ideas are either INFALLIBLE or SUPPORTED. The infallible version was refuted by Hume among others. As a matter of logic, inductive conclusions aren't infallibly proven. It doesn't work. Even if you think deduction or math is infallible (it's not), induction STILL wouldn't be infallible.
Infallible means error is ABSOLUTELY 100% IMPOSSIBLE. It means we'll never improve our idea about this. This is it, this is the final answer, the end, nothing more to learn. It's the end of thinking.
Although most Objectivists (and most people in general) are infallibilists, Objectivism rejects infallibilism. Many people are skeptical of this and often deny being infallibilists. Why? Because they are only infallibilists 1% of the time; most of their thinking, most of the time, doesn't involve infallibilism. But that makes you an infallibilist. It's just like if you only think 1% of haunted houses really have a ghost, you are superstitious.
So suppose induction grants fallible support. We still haven't said how you do induction, btw. But, OK, what does fallible support mean? What does it do? What do you do with it? What good is it?
Support is only meaningful and useful if it helps you differentiate between different ideas. It has to tell you that idea X is better than idea Y which is better than idea Z. Each idea has an amount of support on a continuum and the ones with more support are better.
Apart from this not working in the first place (how much support is assigned to which idea by which induction? there's no answer), it's also irrational. You have these various ideas which contradict each other, and you declare one "better" in some sense without resolving the contradiction. You must deal with the contradiction. If you don't know how to address the contradiction then you don't know which is right. Picking one is arbitrary and irrational.
Maybe X is false and Y is true. You don't know. What does it matter that X has more support?
Why does X have more support anyway? Every single piece of data you have to induce from does not contradict Y. If it did contradict Y, Y would be refuted instead of having some lesser amount of support. Every single piece of data is consistent with both X and Y. It has the same relationship with X and with Y. So why does Y have more support?
So what really happens if you approach this rationally is everything that isn't refuted has exactly the same amount of support. Because it is compatible with exactly the same data set. So really there are only two categories of ideas: refuted and non-refuted. And that isn't induction. I shouldn't have to say this, but I do. That is not induction. That is Popper. That is a rejection of induction. That is something different. If you want to call that "induction" then the word "induction" loses all meaning and there's no word left to refer to the wrong ideas about epistemology.
Why would some piece of data that is consistent with both X and Y support X over Y? There is no answer and never has been. (Unless X and Y are themselves probabilistic theories. If X says that a piece of data is 90% likely and Y says it's 20% likely, then if that data is observed the Bayesians will start gloating. They'd be wrong. That's another story. But why should I tell it? You wouldn't have thought of this objection yourself. You only know about it because I told you, and I'm telling you it's wrong. Anyway, for now just accept that what I'm talking about works with all regular ideas that actually assert things about reality instead of having built-in maybes.)
Also, the idea of support really means AUTHORITY. Induction is one of the many attempts to introduce authority into epistemology.
Authority in epistemology is abused in many ways. For example, some people think their idea has so much authority that if there is a criticism of it, that doesn't matter. It'd take like 5 criticisms to reduce its authority to the point where they might reject it. This is blatantly irrational. If there is a mistake in your idea it's wrong. You can't accept or evade any contradictions, any mistakes. None. Period.
Just the other day a purported Objectivist said he was uncomfortable that if there is one criticism of an idea then that's decisive. He didn't say why. I know why. Because that leaves no room for authority. But I've seen this a hundred times. It's really common.
If no criticism is ever ignored, the authority never actually gets to do anything. Irrationally ignoring criticism is the main purpose of authority in epistemology. Secondary purposes include things like intimidating people into accepting your idea.
But wait, you say, induction is a method of MAKING theories. We still need it for that even if it doesn't grant them support/authority.
Well, is it really a method of making theories? There's a big BLANK OUT in the part of induction where it's supposed to actually tell you what to do to make some theories. What is step one? What is step two? What always fills in this gap is intuition, common sense, and sometimes, for good measure, some fallacies (like that correlation implies or hints at causation).
In other words, induction means think of theories however (varies from person to person), call it "induction", and never consider or examine or criticize or improve your methods of thinking (since you claim to be using a standard method, no introspection is necessary).
For any set of data, infinitely many general conclusions are logically compatible. Many people try to deny this. As a matter of logic they are just wrong. (Some then start attacking logic itself and have the audacity to call themselves Objectivists). Should I go into this? Should I give an example? If I give an example, everyone will think the example is STUPID. It will be. So what? Logic doesn't care what sounds dumb. And I said infinitely many general conclusions, not infinitely many general conclusions that are wise. Of course most of them are dumb ideas.
So now a lot of people are thinking: induce whichever one isn't dumb. Not the dumb ones. That's how you pick.
Well, OK, and how do you decide what's dumb? That takes thinking. So in order to do induction (as it's just been redefined), in one of the steps, you have to think. That means we don't think by induction. Thinking is a prerequisite for induction (as just redefined), so induction can't be part of thinking.
What happens here is the entirety of non-inductivist epistemology is inserted as one of the steps of induction and is the only reason it works. All the induction stuff is unnecessary and unhelpful. Pick good ideas instead of dumb ones? We could have figured that out without induction, it's not really helping.
Some people will persevere. They will claim that it's OBVIOUS which ideas are dumb or not – no thinking required. What does that mean? It means they can figure it out in under 3 seconds. This is silly. Under 3 seconds of thinking is still thinking.
Do you see what I mean about there are so many things wrong with induction it's hard to figure out where to start? And it's hard to go through them in an orderly progression because you start talking about something and there's two more things wrong in the middle. And here I am on this digression because most defenses of induction – seriously this is the standard among non-professionals – involve a denial of logic.
So backing up, supposedly induction helps us make theories. How? Which ones? By what steps do we do it? No answers. And how am I supposed to prove a negative? How do I write an essay saying "induction has no answers"? People will say I'm ignorant and if only I read the right book I'd see the answer. People will say that just because we don't know the answer doesn't mean there isn't one. (And remember that refutation of induction I mentioned up top? Remember Popper's arguments that induction is impossible? They won't have read any of that, let alone refuted it.)
And I haven't even mentioned some of the severe flaws in induction. Induction as originally intended – and it's still there but it varies, some people don't do this or aren't attached to it – meant you actually read the book of nature. You get rid of all your prejudices and biases and empty your mind and then you read the answers straight FROM the observation data. Sound like a bad joke? Well, OK, but it's an actual method of how to do induction. It has instructions and steps you could follow, rather than evasion. If you think it's a bad joke, how much better is it to replace those concrete steps with vagueness and evasion?
Many more subtle versions of this way of thinking are still popular today. The idea of emptying your mind and then surely you'll see the truth isn't so popular. But the idea that data can hint or lead or point is still popular. But completely false. Observation data is inactive and passive. Further, there's so much of it. Human thinking is always selective and active. You decide which data to focus on, and which ways to approach the issue, and what issues to care about, and so on. Data has to be interpreted, by you, and then it is you interpretations, not the data itself, which may give you hints or leads. To the extent data seems to guide you, it's always because you added guidance into the data first. It isn't there in the raw data.
Popper was giving a lecture and at the start he said, "Observe!" People said, "Observe what?" There is no such thing as emptying your mind and just observing and being guided by the data. First you must think, first you must have ideas about what you're looking for. You need interests, problems, expectations, ideas. Then you can observe and look for relevant data.
The idea that we learn FROM observation is flawed in another way. It's not just that thinking comes first (which btw again means we can't think by induction since we have to think BEFORE we have useful data). It also misstates the role of data in thinking. Observations can contradict things (via arguments, not actually directly). They can rule things out. If the role of data is to rule things out, then whatever positive ideas we have we didn't learn from the data. What we learned from the data, in any sense, is which things to reject, not which to accept.
Final point. Imagine a graph with a bunch of dots on it. Those are data points. And imagine a line connecting the dots would be a theory that explained them. This is a metaphor. Say there are a hundred points. How many ways can you draw a line connecting them? Answer: infinitely many. If you don't get that, think about it. You could take a detour anywhere on the coordinate plane between any two connections.
So we have this graph and we're connecting the dots. Induction says: connect the dots and what you get is supported, it's a good theory. How do I connect them? It doesn't say. How do people do it? They will draw a straight line, or something close to that, or make it so you get a picture of a cow, or whatever else seems intuitive or obvious to them. They will use common sense or something – and never figure out the details of how that works and whether they are philosophically defensible and so on.
People will just draw using unstated theories about which types of lines to prefer. That's not a method of thinking, it's a method of not thinking.
They will rationalize it. They may say they drew the most "simple" line and that's Occam's razor. When confronted with the fact that other people have different intuitions about what lines look simple, they will evade or attack those people. But they've forgotten that we're trying to explain how to think in the first place. If understanding Occam's razor and simplicity and stuff is a part of induction and thinking, then it has to be done without induction. So all this understanding and stuff has to come prior to induction. So really the conclusion is we don't think by induction, we have a whole method of thinking which works and is a prerequisite for induction. Induction wouldn't solve epistemology, it'd presuppose epistemology.
What we really know, from the graph with the data points, is that all lines which don't go through every point are wrong. We rule out a lot. (Yes, there's always the possibility of our data having errors. That's a big topic I'm not going to go into. Regardless, the possibility of data errors does not help induction's case!)
And what about the many lines which aren't ruled out by the data? That's where philosophy comes in! We don't and can't learn everything from the data. Data is useful but isn't the answer. We always have to think and do philosophy to learn. We need criticisms. Yes, lots of those lines are "dumb". There are things wrong with them. We can use criticism to rule them out.
And then people will start telling me how inconvenient and roundabout that is. But it's the only way that works. And it's not inconvenient. Since it's the only way that works, it's what you do when you think successfully. Do you find thinking inconvenient? No? Then apparently you can do critical thinking in a convenient, intuitive, fast way.
At least you can do critical thinking when you're not irrational defending "induction" because in your mind it has authority.
EDIT: Some more problems with induction I didn't mention:
"the future is like the past" principle. or "the future will resemble the past". or other similar formulations. the problem is the future always resembles the past in some ways and not others. which are which? induction doesn't tell you which are which. instead it implicitly smuggles in thinking as a prerequisite to figure that out. (so it presupposes, rather than explains, thinking).
Induction is supposed to be a solution to epistemology but doesn't address Paley's problem.
The issue of how we think about abstract non-empirical ideas is not addressed by induction.
See also: http://www.curi.us/1450-new-induction-disproof