Now, I can understand what it means to observe something circular in nature, but to what are you referring when you say that they “formulated an explanation”? Formulating explanations is integral to the scientific method, but what kind of a causal explanation are you proposing to give that is relevant to grasping the concept of circularity? What kinds of predictions would you make and how exactly would you test them?

By that, I mean that they came up with a way to explain the concept of a circle to someone else. Without living back then, I can't know exactly what they did, but whatever they did was intended to demonstrate the concept to others, which serves the same purpose as explaining it in causal terms. It probably worked something like this:

- Observe something that is naturally circular.
- Describe it in such a way that it isn't confused with something that is similarly shaped but not circular.
- Predict that other circular objects will conform to that description.
- Confirm by checking other circular objects against the description formulated, changing it if need be.

I think that you are mistakenly equating the first step in the scientific method (observation) with the scientific method itself.

This is your opinion, and it's based on your incomplete understanding of what I said - as you stated just above, you didn't know what I was referring to by "formulated an explanation", but instead of waiting for me to answer, you tried to imagine what I might have meant. As a result, you came up with an idea that was only tangentially related to what I was trying to get across, and ultimately ended up being incorrect.

The intuitive grasp that you and I have of what the abstract concept of circularity means is, along with other geometrical, mathematical, and logical concepts, what grounds our ability to reason scientifically. You can certainly say that one must observe something that is roughly circular before conceiving of the definition of a circle, but that doesn’t mean that that observation is also sufficient to generate understanding of what a circle is. To come to an understanding of what a circle is, one has to make an ‘intuitive leap’; or as you aptly put it, ‘conceive of’ it.

And just what do you think the "abstract concept of circularity" is, if it is not an explanation that can be tested against circle-like objects to tell whether they are circular or not? And in the process, occasionally come up with a way to more accurately describe the concept due to that very testing?? Which, notably, is how the scientific method works. It doesn't matter if the initial explanation comes as a result of reasoning or due to intuition as long as it can be communicated to others and tested against reality. And either way, it still comes about as a result of an observation.

To put it another way, consider how the authors of geometry and physics textbooks respectively present their discipline’s knowledge. If a physicist wants to explain atomic theory to a class she will likely present a process. She will mention the names of people like Michael Faraday, J.J. Thomson, and Ernest Rutherford; she will detail the many experiments that these scientists did; and she will talk about the various hypotheses that were proposed and tested (e.g. the ‘plum pudding model’ versus the ‘nuclear atom model’). In contrast, if you read a geometry textbook, the authors simply present the basic definitions and axioms as fact and proceed from there; the student is expected to intuitively grasp the basic concepts as they are presented. Of course, if I am wrong then you should be able to pick up a geometry textbook and read about the history of the scientific process that was used to obtain the definition of a circle that we use today.

This is because basic geometric concepts (such as the formula for a circle) were worked out so long ago that we have no idea who came up with them. Furthermore, you might note that later geometric concepts, which are not so basic, are credited to the person who discovered them, as well as the proofs they used to demonstrate the accuracy of the concept. I won't deny that math works differently than various sciences, but to claim that students are simply expected to 'intuitively' grasp basic mathematical concepts is fallacious. If it were a simple matter of intuition, we wouldn't have to use various geometrical formulas, and we wouldn't have to teach children math. We would be able to grasp those formulas 'intuitively', just as children would 'intuitively' grasp even more basic mathematical concepts (such as addition, subtraction, multiplication, division, fractions...).

Furthermore, nobody needs to see more than one object that approximates circularity to understand the abstract concept of circularity; we simply grasp the truth intuitively and no number of repeated observations changes or strengthens our conception of what a circle is.

No, we don't "grasp the truth", because math isn't a matter of 'truth' to begin with. What we grasp is a

**concept** - but one that we can strengthen through repeated observation. This is easily confirmed by observing the way that young children learn language. When they grasp a concept, they start applying that concept to anything that comes close to matching it - for example, calling a cat or a mouse a 'doggy' because they grasped the concept of a furry animal as being a 'doggy', but haven't learned to differentiate between different kinds of furry animals yet.

If, however, you are correct when you say that people “made changes as needed in order to make sure it [the definition of a circle] conformed to reality” then you have a rather paradoxical result on your hands. The assumption that the definition of a circle is tentative (as all scientific theories are to some degree or another) would mean that our confidence in its correctness would depend upon finding examples of it in the real world. The more objects we find that fit the definition the greater confidence we will have in its correctness, but if the opposite is true and we find few or no objects that meet the criteria then we will have to make an ‘adjustment’.

Here is another misunderstanding you have, this time about science. Many scientific theories are not 'tentative' in the way that you mean. While they may start out as 'tentative', as they are tested, they become progressively less so. While it's possible that a theory that has been well-tested may be found to be incorrect in some way, it is most likely that this incorrectness is in something that nobody thought to test, or that nobody could test. Much like how Newton's classical model is accurate except under certain circumstances (approaching the speed of light, approaching a singularity, neither of which he could realistically test), and thus was incorporated into Einstein's own theory.

Unfortunately, since there are no objects in nature (at least none that I am aware of) that perfectly conform to the definition of circularity, it would seem that actually applying the scientific method to the definition of a circle would decrease rather than increase our certainty in its correctness. So it seems to me that someone like Aristotle would have been more justified in using the definition of a circle that I proposed since, for all he knew, the things in nature that looked circular actually were perfect circles; nowadays, with our electron scanning microscopes and our understanding of microstructure, we know better.

Why would it? We don't have to have a perfectly circular object in order to determine what a circle is. And if we find something that more perfectly describes a circle than what we already have, why would we not improve our definition of a circle by incorporating it? In other words, much as science works. Now, it's true that it unlikely that we'll find a better formula to describe a circle than what we already have, but there are plenty of mathematical formulas that can be improved. Like, say, the value of pi.

By the way, isn't pi incorporated into the formulas which are part of the definition of a circle? And wouldn't that mean that as we more accurately determine the value of pi, that we also can more accurately calculate those same formulas?

Regarding opinions, you gave the example of someone saying that they “like chocolate”. I think that it is important to note that words as they are used in popular parlance do not always mean the same thing to a philosopher or a scientist. An example would be the word ‘theory’ which creationists often use derisively to refer to evolution as ‘only a theory’ despite the fact that scientists mean something quite different when they use the word.

Granted, mainly to get this out of the way.

In the same way, philosophers are not referring to trivial flavour preferences when they use a word like ‘intuition’; rather, they are referring to statements like the law of the excluded middle or the first premise to the kalam cosmological argument or to moral intuitions like “it is wrong to torture babies for fun”. Concepts like these intuitively seem to be true and ground all of our scientific and metaphysical reasoning.

To address these one by one:

The law of the excluded middle is not an 'intuition', nor are the other two classical laws of thought. Indeed, they are actually not particularly intuitive, in and of themselves. It would be better to call them instinctive, similar to language. Which is to say that we are biologically wired to incorporate them without having to think about it.

As for the Kalam cosmological argument and others of its ilk, they are not necessarily true simply because they are intuitive. For example, the first premise states that there must have been a first cause to the universe (because of the cause-effect chain), but we are finding that this may not have actually been the case (for example, some quantum effects are not 'caused', they simply happen). So in this case, our intuition (which is based on our experiences here on Earth) is quite possibly wrong.

And finally, moral 'intuitions' are actually instinctively-learned rules of the culture one is raised in. If there are any universal morals, they are only those which are necessary for a society/culture to survive. Other than that, all bets are off.

My point is that the person who wants to study quantum mechanics or evolution or any other scientific project utilizes the same kinds of intuitive background assumptions as someone who wishes to pursue the project of natural theology. As such, if it is reasonable to accept knowledge gained through the scientific method, then it is also legitimate to accept knowledge gained through the arguments of natural theology.

This is logically flawed. It is like saying that because you use the same materials for two buildings, that both are structurally sound - without considering any other aspects of the buildings. On top of that, there is the problem that logic is only as sound as its premise. If you start from a false premise, then no matter how good your logic is, you're going to end up with a wrong answer. And these "intuitive background assumptions" you talk about are not the premise of an argument. In other words, your argument here is wrong. You cannot automatically legitimize information gained through natural theology simply because you can gain information through the scientific method using similar basic assumptions.

You claimed that rational introspection, like opinions, it is useless for obtaining knowledge since its suppositions cannot be proven.

Actually, no, I said that you can't disprove a hypothesis with it. That means rational introspection is useless for gaining knowledge by itself, because you can't filter out the bad data from the good. You have to test information (no matter how you gain it) against reality to determine if it is useful knowledge.

Interestingly enough, in the scientific world it is not really ‘provability’ that scientists strive for but rather falsifiability. Physicists don’t say that Einstein’s theory of quantum mechanics has been proven therefore we can move on to other things; rather, they point out the possible ways that his theory could be falsified - it hasn’t happened yet, but it could since a superior theory could become available in the future.

I am well aware of that. It's why I said you couldn't disprove something with rational introspection, rather than that you couldn't prove it.

You can certainly say that metaphysical intuitions cannot be proven, but that doesn’t mean that falsification is impossible. For instance, when someone says that ‘everything that begins to exist must have a cause’ that statement, if false, is open to counterexamples. A couple of legitimate counter examples would certainly serve to undermine my confidence in the intuitive plausibility of the claim. In contrast, someone’s opinion about chocolate isn’t amenable to being proven or falsified in the sense in which we are talking here, and is therefore useless as a grounding premise for gaining knowledge.

There are counter-examples (albeit not proven yet). For example, the virtual particles that cause Hawking radiation if they occur next to a black hole are not 'caused'.

And in any case, it's been pretty well demonstrated that intuition is not reliable in many respects. For example, in the field of probability. For example, in the famous birthday problem, our intuition tells us that in order to have a 50% probability of a matched birthday, you need a large percentage of the total birthdays available to check. In fact, you only need 23 randomly-picked people in a room to have a 50% chance of a birthday match - less than 10% of the total number. Completely counter-intuitive.