tag:blogger.com,1999:blog-2492055523235356445.post8548304042836890295..comments2020-01-12T03:55:47.083-08:00Comments on - the dance of reason: When 2+2 Does Not Equal 4Sac State Philosophyhttp://www.blogger.com/profile/17963066908030437925noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-2492055523235356445.post-3334947899220925862018-12-05T12:18:41.640-08:002018-12-05T12:18:41.640-08:00Hi John, thanks for the feedback.
In re (1): I th...Hi John, thanks for the feedback.<br /><br />In re (1): I think there are some difficult questions about categories, what they are, how to think about them, etc. Obviously Plato and Aristotle are going to be big dogs in that fight, but I don't have much sympathy for either of their approaches. Russell might take a more set-theoretic approach, but I have issues with that, too. <br /><br />In short, I can't claim to have a general theory of categories that squares with my naturalistic thinking. But as far as the math goes, categories might be one of those instances (of which there are many) when 2+2 really does equal 4. It's only in very rare circumstances where that doesn't hold true.<br /><br />(2) I think Boyd's approach is an interesting one, but I'm not sure it will resolve the problem. To use another example: if you add two cups of liquid water to two cups of frozen water you'll get four cups... until either the frozen water melts (in which case you'll have less than four cups) or the liquid water freezes (in which case you'll have more than four cups). <br /><br />Now, does Boyd's theory treat liquid water and frozen water as the same 'natural kind' or different ones? If they're the same, his account will have trouble with the end-state (we added 2+2 and got more/less than four). If they're different then he has problems with the beginning-state (we added two of one kind to two of another kind, but nonetheless got four of the same kind.) <br /><br />Of course, these 'problems' are only problems if Boyd wants to resist my conclusion, that 2+2 doesn't always equal 4. If he's willing to accept that, then these aren't problems, that's just the way the world works sometimes.Garret Merriamhttps://www.blogger.com/profile/08572878516975431059noreply@blogger.comtag:blogger.com,1999:blog-2492055523235356445.post-22586385486596849392018-12-05T12:02:28.966-08:002018-12-05T12:02:28.966-08:00Hi Randy, thanks for the feedback. Yes, I think we...Hi Randy, thanks for the feedback. Yes, I think we're generally in agreement here. I was tempted to speak more directly to my own Quineian sensibilities, but I ran out of space. I'm thinking about appealing to him more directly in another post on the recent redefinition of the kilogram. But that's for next semester. <br /><br />I probably could have been more clear about other, non-Platonistic forms of mathematical realism, but again I felt pressed on the word count. But I hadn't thought about the issue from Peano's perspective, that's an interesting take. I can't say I have any clear intuitions on that approach, but I'd love to hear more about it. Garret Merriamhttps://www.blogger.com/profile/08572878516975431059noreply@blogger.comtag:blogger.com,1999:blog-2492055523235356445.post-41121855407502764142018-12-04T13:10:28.876-08:002018-12-04T13:10:28.876-08:00I'm also a naturalist on numbers. However, he...I'm also a naturalist on numbers. However, here are some thoughts: 1) What if someone says 1+1=2 when we're adding categories? Instead of adding particular things in the real world like cups of water or hair, one can jump up to a higher level of abstraction to superordinate categories. For example, if I add the category of hammers with the category of pencils, there will be 2 categories. <br /><br />2) Regarding whether we can say that 2 cups of water are the same thing despite some differences in trace minerals and atoms: What if we take something like Richard Boyd's homeostatic cluster theory of natural kinds? Water has properties like being a clear, odorless, tasteless, liquid that fills rivers and streams. These are not necessary and sufficient conditions. These properties are likely, however, to cluster together given an underlying chemical causal mechanism. There can be some disparate atoms in the different cups but as long as the core chemical mechanism is present, then that is fine. We use the cluster of properties and the underlying causal mechanism to determine that both cups of water are cups of water. Since the cluster of properties are not nec/suff conditions for classification, this also allows for some differences in the properties between both cups. In this fashion, we can conclude that both cups are the same in the sense that they both can be classified as cups of water.John J. Parkhttps://www.blogger.com/profile/03000310555845080326noreply@blogger.comtag:blogger.com,1999:blog-2492055523235356445.post-70825394925289617442018-12-04T09:29:52.213-08:002018-12-04T09:29:52.213-08:00Garret, thanks for this. It's an important, fu...Garret, thanks for this. It's an important, fundamental issue for students to be able to think through. It also connects really well with my previous post on spoilers and keepers.<br /><br />I'm pretty much in your camp with respect to your naturalistic considerations, as you know. I also pretty much accept Quine's characterization of analyticity, not as something that holds true no matter what, but as something that, like probability, admits of degrees. Basically, the less susceptible a proposition is to empirical disconfirmation, the more analytic it is. So, in this sense, 2 + 2 = 4 can remain an excellent example of an analytic truth. (Of course, the sense of 'plus' you use when when talking about concatenating or collecting physical objects is not, and never has been, regarded as analytic, which I think you mean to acknowledge.) When you go at analyticity in this way, then you get the same practical result as you get with probability. i.e., just as .999999999999 probable is certain for all practical purposes, .999999999999 analytic is true no matter what for all practical purposes. <br /><br />This connects to my previous posts in the following way: When we do philosophy, we are not simply in the business of determining whether there is any such thing as X, in this case analytic truth. We are really in the business of determining whether a particular definition of X is useful, and whether a different one might be better. If we suspect the latter, we propose an explication of X. Quine did not argue that analyticity does not exist. He explicated the concept in a way that made it more epistemically defensible and useful. <br /><br />Another small point: I'm sure you would want your readers to understand that Platonism isn't the only option for someone who wants to preserve the analyticity of 2 + 2 = 4. Mathematical equations are, in the end, just shorthand for counting. On Peano's axioms, their analyticity reduces to the claim that 0 is a number and that every number has a successor. So if you want to attack the analyticity of 2 + 2 = 4, you really need to be arguing that 0 might not be a number and the successor of 0 might not be 1, both of which are axioms in Peano’s system and generally held to be true by definition.G. Randolph Mayeshttps://www.blogger.com/profile/18285281186698499962noreply@blogger.com