(are the lines between the arrowheads of the same or of different length? how do you know?)
A re-post from Facebook:
I buy a package that has 12 beer bottles in it (it says so on the package, and for a number of reasons, I happen to believe the label; also, maybe I can feel the bottles). I tear a side of the package and take out a bottle. My friend takes out another bottle.
Without looking inside the package, feeling it, etc., I KNOW that there are 10 bottles left in the package. This fact is independent of the type of beer, of the colors of the bottles, or of any other properties. In fact, this observation is true even regardless of the kinds of objects I am counting.
This observation is not empirical. I.e., one might say that I have encountered the "12–2" situation enough times to know that there are 10 items left. The same way that from experience, if I hear a certain noise from my engine, I know what's wrong with it without looking inside (or if I drop the beer pack from some height and hear a certain noise, I know that at least one bottle broke inside).
I can imagine the situation or a universe (or a kind of engine) that would make a certain noise that did not indicate that kind of breakage. I can imagine a universe in which when I dropped an object from a great height it wouldn't break. I cannot imagine a situation in which 12–2 would not equal 10. It's a necessity of reality.
But it's not an introspective subjective observation either. It's not the same as me liking fish and my wife hating it. First of all, I expect both my wife and myself to agree on 12–2. Second, I think the internal knowledge that there are 10 bottles left is about an objective fact outside my head (assuming the bottles' existence is also an objective fact and not an illusion, which I am inclined to believe).
So, what is this knowledge? Where does it come from, and what does it mean?
Some say that it is knowledge of properties of groups objects (their number) that comes from intuition. But that doesn't seem right because we have to be taught addition and subtraction.
It might be that numerical truths are objective truths about the world, basics of which we acquire empirically (for instance, we know that items come in groups, and we can count them using our fingers), but relationships between which we can know logically (I have never in my life verified that 4 groups of apples, each containing 23 apples, makes up the total of 92 apples, but from my knowledge of basic numeric properties of objects, I can deduce that).
Anyway, some people say the same about moral truths: that they are objective properties of some objects and events that we know about intuitively (perhaps we need to learn the basics of them from experience, such as by trying certain events — for instance, those producing pain — on ourselves, but once we have learned the foundational moral truths, we can deduce more complicated moral assumptions logically and intuitively).
We know that murder is wrong; it is wrong in all situations, between all kinds of beings. (There may be nuances which may make the act not murder; for instance, self defense, killing of passion, human sacrifice, war, etc. — we may argue which of these are murder, but once we agree that, for instance, killing of passion is always murder, we will necessarily agree that it is wrong. This is similar to asking "how many objects are there in the package" and disagreeing whether we count bottles and caps as separate objects. Once we agree that, for instance, they are one object, we will also necessarily agree on their numerical properties: such that, if we took out two of them, there must be ten left.)
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The same way, people get logical illusions about numbers; for instance, in statistics. But they can be shown, not empirically, but logically, that they are wrong. And the same can be done with ethics: people may believe that inequality is bad, but one might be able to show them that it is not (by presenting different scenarios of inequality that even they would find morally satisfactory, while forced equality under those scenarios would be in fact bad), and what they think is bad is something else (poverty, aggression of the mighty over the weak, etc.).
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An important logical/philosophical argument against anti-intuitionism is that all of our rational facts, rules, observations, and deductions, must eventually rest on some set of intuitions. For instance, my intuition that Dr. X is a good physician may be wrong if it's based only on his charisma. But imagine that I define "good" as "having adequate knowledge and skills to diagnose a sickness with a high [insert whatever number if you want] probability of success". Then I verify that this is so experimentally. Or I know that someone else has verified so. Or I trust his medical certification. Etc.
Well, in each of those deductions, I have relied on a number of intuitions: namely, that my observations and my logic are trustworthy. There is a good (existential) reason to rely on those intuitions (and believe that they are more trustworthy than my intuition based on the doctor's charisma), but they remain intuitions nonetheless.
What's the alternative? That every rational concept must be derived from another rational concept which is derived from another rational concept — and so on, an infinite regress of derivations. Or that the derivations are circular. Both seem to be less satisfactory (to put it mildly) than accepting the fact that all our knowledge rests eventually on a set of intuitions.
An audio of an excerpt from Dr. Michael Huemer's book: