P1. \u00a0An actual infinite cannot exist.<\/p>\n
P2. \u00a0An infinite temporal regress of events is an actual infinite.<\/p>\n
P3. \u00a0Therefore, an infinite temporal regress of events cannot exist.<\/p>\n
I’ve already talked about this argument in a few places:<\/p>\n
Infinite dreams<\/span><\/a><\/p>\n Infinity minus infinity<\/a><\/p>\n Can God count to infinity?<\/a><\/p>\n And a few more posts will be devoted to this topic. \u00a0But here I’ll consider another angle.<\/p>\n In defending their argument, Craig and Sinclair consider several possible refutations to\u00a0the premise P1. \u00a0One of these is that the existence of infinitely many numbers (and other kinds of mathematical objects) might directly contradict P1.<\/p>\n As Craig and Sinclair observe, this raises a very interesting question. \u00a0In what sense can it be said that mathematical objects like numbers\u00a0exist?<\/em> \u00a0Do they exist as material objects, perhaps only as configurations of neurons in the brains of humans? \u00a0Do they have some kind of disembodied existence in a platonic mathematical realm? \u00a0Or is there really no meaningful sense in which numbers can be said to exist? \u00a0Is there some other explanation?<\/p>\n As a mathematician, I find these questions fascinating, though I certainly don’t claim to know the answers. \u00a0I’ve talked about these ideas with several of my colleagues over the years, and discovered that there are a range of views, and also a range of associated confidence levels.\u00a0\u00a0The following diagram from the chapter illustrates several different kinds of theories of mathematical ontology (ie, theories about the existence of mathematical objects):<\/p>\n There are certainly a lot of views! \u00a0To me, none of them seem likely to be provable – at least not in a strict sense – and I think the range of views held by professional mathematicians and philosophers counts as evidence that there does not currently exist a definitive proof one way or another. \u00a0With that said, though, I do have doubts about a few of them. \u00a0Some people have argued that we shouldn’t accept anything like a platonist<\/a> view, for example, since there is no evidence that platonic objects exist. \u00a0But,\u00a0according to a recent survey, 39.3% of professional philosophers are platonists<\/a> (with 37.7% nominalists and 23.0% choosing some other option). \u00a0Most arguments I’ve heard seem to suggest that agnosticism is a reasonable position to take on the matter.<\/p>\n Now, returning to Craig and Sinclair’s argument, we see that consideration of mathematical objects only proves to be a decisive refutation<\/em> to P1 given certain realist views of mathematical ontology. \u00a0Here is how Craig and Sinclair put it:<\/p>\n “The Realist, then, if he is to maintain that mathematical objects\u00a0furnish a decisive counterexample to the denial of the existence of the actual infinite,\u00a0must provide some overriding argument for the reality of mathematical objects, as well as\u00a0rebutting defeaters of all the alternatives consistent with classical mathematics \u2013 a task\u00a0whose prospects for success are dim, indeed. It is therefore open to the [apologist] to hold\u00a0that while the actual infinite is a fruitful and consistent concept within the postulated\u00a0universe of discourse, it cannot be transposed into the real world.”<\/p>\n [As an aside, it seems that Craig and Sinclair are urging an insider of the Mathematical Realist camp to rationally assess his beliefs, and subject them to the same level of scrutiny he might demand of others. \u00a0Is this an Outsider Test for Mathematical Ontology?]<\/p><\/blockquote>\n The astute reader will have noticed that Craig and Sinclair are attempting to shift their burden of proof here. \u00a0They are making the argument. \u00a0And they must offer support for their premises. \u00a0If they are aware that the existence of mathematical objects could cast doubt on one of their key premises, then they must adequately deal with this possibility. \u00a0It is not enough to simply say that they think it would be difficult for an opponent to conclusively refute the premise.<\/p>\n It is true\u00a0that to conclusively\u00a0refute\u00a0P1 by considerations of mathematical objects, one must demonstrate the validity of a theory of mathematical ontology that involves the actual existence of mathematical objects.<\/p>\n But, on the other hand, to conclusively prove P1, Craig and Sinclair (or someone else) must demonstrate\u00a0the validity of a theory of mathematical ontology that involves the actual non-existence of mathematical objects – which I think Craig and Sinclair would probably agree is also “a task\u00a0whose prospects for success are dim, indeed”. \u00a0If they do not successfully do this, we are left wondering if perhaps mathematical objects do\u00a0<\/em>provide a counterexample to P1. \u00a0But how might one prove such an assertion? \u00a0Proving the non-existence of mathematical objects seems akin to proving the non-existence of god(s)!<\/p>\n In conclusion, I don’t know which theory of mathematical ontology is the correct one, and I don’t think Craig and Sinclair do either. \u00a0If some form of platonism is correct (as is believed by nearly 40% of professional philosophers), mathematical objects would indeed provide a decisive counterexample to P1. \u00a0And, unless this possibility can be ruled out, we have no compulsion to accept the premise – or the conclusion of any argument that relies on it.<\/p>\n","protected":false},"excerpt":{"rendered":" William Lane Craig and James Sinclair’s chapter on the Kalam Cosmological Argument\u00a0in the Blackwell Companion to Natural Theology\u00a0contains a…<\/p>\n","protected":false},"author":16,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8,14,52,17,25],"tags":[193,197,199,217,202,215,216,209],"class_list":["post-1300","post","type-post","status-publish","format-standard","hentry","category-cosmological-argument","category-infinity","category-logic","category-mathematics","category-william-lane-craig","tag-cosmological-argument","tag-god","tag-infinity","tag-logic","tag-mathematics","tag-philosophy","tag-theism","tag-william-lane-craig"],"_links":{"self":[{"href":"https:\/\/skepticink.com\/reasonablyfaithless\/wp-json\/wp\/v2\/posts\/1300","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/skepticink.com\/reasonablyfaithless\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/skepticink.com\/reasonablyfaithless\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/skepticink.com\/reasonablyfaithless\/wp-json\/wp\/v2\/users\/16"}],"replies":[{"embeddable":true,"href":"https:\/\/skepticink.com\/reasonablyfaithless\/wp-json\/wp\/v2\/comments?post=1300"}],"version-history":[{"count":0,"href":"https:\/\/skepticink.com\/reasonablyfaithless\/wp-json\/wp\/v2\/posts\/1300\/revisions"}],"wp:attachment":[{"href":"https:\/\/skepticink.com\/reasonablyfaithless\/wp-json\/wp\/v2\/media?parent=1300"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/skepticink.com\/reasonablyfaithless\/wp-json\/wp\/v2\/categories?post=1300"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/skepticink.com\/reasonablyfaithless\/wp-json\/wp\/v2\/tags?post=1300"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"mathematical](\"https:\/\/skepticink.com\/reasonablyfaithless\/files\/2013\/05\/mathematical-objects.jpg\")
<\/a><\/p>\n