No. But that is essentially true by definition.
Sure, but it's not true by definition that numbers are abstract. Given your analogy to mass and velocity, and your view that mathematical objects help explain the unreasonable effectiveness of mathematics, it seems to me that it would make much more sense to treat these number things as playing a causal or constitutive role in the makeup of our universe itself, e.g., as universals. Then it would no longer just be a coincidence that our world conveniently accompanies a causally dislocated Realm of correlates for our mathematical discourse.
To me this is like hearing "If mass and velocity didn't exist Newtonian physics would be just as 'unreasonably effected'.
But it makes a difference to how our world is that objects have velocity and mass. By hypothesis, it doesn't make a difference to how our world is that there are numbers. (And from this it follows that it wouldn't make a difference if there weren't numbers.) If numbers do play a role as worldly 'difference-makers' of some special sort, then could you explain more clearly what that role is, since it's not causal?
Mathematical objects are part of mathematics.
I don't know what that means. If by 'mathematics' you have in mind a set of human behaviors or mental states, then mathematics isn't abstract, so its objects are neither causally nor constitutively in any relation to it. On the other hand, if by 'mathematics' you have in mind another abstract object, then your statement may be true, but I don't see the explanatory relevance to mathematical practice.
The fact that math is unreasonably effective is why we can say mathematical facts are true and mathematical entities exist.
Sure, but it's also why we can assert doctrines like mathematical fictionalism and nominalism. A condition for saying anything at all is that our world exhibit the basic features (property repetition, spatiotemporal structure...) that suffice for there to be worldly quantities at all. I can make sense of the idea that we need to posit something number-like to account in some causality-like way for things like property repetition and spatiotemporal structure themselves. But I still haven't wrapped my head around why assuming numbers are not difference-makers for the physical world (unlike the presence of e.g. velocity), we should posit them to explain the efficacy of theories whose efficacy they have no impact upon.
Just like the fact that quantum theory is unreasonably effective is the reason we can say that quarks exist.
The properties of quarks causally impact our quantum theorizing. In a world where there weren't quarks, we'd be less likely to have the evidence for them that we do. If that isn't true of mathematics (or, in some ways even worse, if we can't even coherently talk about 'mathless worlds'), then I don't see the parity.
Sure, but it's not true by definition that numbers are abstract.
Huh?
it seems to me that it would make much more sense to treat these number things as playing a causal or constitutive role in the makeup of our universe itself, e.g., as universals.
I don't recognize a difference between universals and abstract objects but neither plays a causal role in the make up of the universe.
Then it would no longer just be a coincidence that our world conveniently accompanies a causally dislocated Realm of correlates for our mathematical discourse.
You're taki...
LessWrong has twice discussed the PhilPapers Survey of professional philosophers' views on thirty controversies in their fields — in early 2011 and, more intensively, in late 2012. We've also been having some lively debates, prompted by LukeProg, about the general value of contemporary philosophical assumptions and methods. It would be swell to test some of our intuitions about how philosophers go wrong (and right) by looking closely at the aggregate output and conduct of philosophers, but relevant data is hard to come by.
Fortunately, Davids Chalmers and Bourget have done a lot of the work for us. They released a paper summarizing the PhilPapers Survey results two days ago, identifying, by factor analysis, seven major components consolidating correlations between philosophical positions, influences, areas of expertise, etc.
Anti-Naturalists tend to work in philosophy of religion (.3) or Greek philosophy (.11). They avoid philosophy of mind (-.17) and cognitive science (-.18) like the plague. They hate Hume (-.14), Lewis (-.13), Quine (-.12), analytic philosophy (-.14), and being from Australasia (-.11). They love Plato (.13), Aristotle (.12), and Leibniz (.1).
A disproportionate number of objectivists work in normative ethics (.12), Greek philosophy (.1), or philosophy of religion (.1). They don't work in philosophy of science (-.13) or biology (-.13), and aren't continentalists (-.12) or Europeans (-.14). Their favorite philosopher is Plato (.1), least favorites Hume (-.2) and Carnap (-.12).
Rationalists are wont to work in metaphysics (.14), and to avoid thinking about the sciences of life (-.14) or cognition (-.1). They are extremely male (.15), inordinately British (.12), and prize Frege (.18) and Kant (.12). They absolutely despise Quine (-.28, the largest correlation for a philosopher), and aren't fond of Hume (-.12) or Mill (-.11) either.
They are extremely female (.17) and young (.15 correlation coefficient for year of birth). They work in ethics (.16), social/political philosophy (.16), and 17th-19th century philosophy (.11), avoiding metaphysics (-.2) and the philosophies of mind (-.15) and language (-.14). Their heroes are Kant (.23), Rawls (.14), and, interestingly, Hume (.11). They avoid analytic philosophy even more than the anti-naturalists do (-.17), and aren't fond of Russell (-.11).
5. Externalists: Really, they just like everything that anyone calls 'externalism'. They think the content of our mental lives in general (.66) and perception in particular (.55), and the justification for our beliefs (.64), all depend significantly on the world outside our heads. They also think that you can fully understand a moral imperative without being at all motivated to obey it (.5).
Trekophobes are relatively old (-.1) and American (.13 affiliation). They are quite rare in Australia and Asia (-.18 affiliation). They're fairly evenly distributed across philosophical fields, and tend to avoid weirdo intuitions-violating naturalists — Lewis (-.13), Hume (-.12), analytic philosophers generally (-.11).
They're also fond of a wide variety of relatively uncontroversial, middle-of-the-road views most philosophers agree about or treat as 'the default' — political egalitarianism (.33), abstract object realism (.3), and atheism (.27). They tend to think zombies are metaphysically possible (.26) and to reject personal identity reductionism (.26) — which aren't metaphysically innocent or uncontroversial positions, but, again, do seem to be remarkably straightforward and banal approaches to all these problems. Notice that a lot of these positions are intuitive and 'obvious' in isolation, but that they don't converge upon any coherent world-view or consistent methodology. They clearly aren't hard-nosed philosophical conservatives like the Anti-Naturalists, Objectivists, Rationalists, and Trekophobes, but they also clearly aren't upstart radicals like the Externalists (on the analytic side) or the Anti-Realists (on the continental side). They're just kind of, well... obvious.
Conventionalists are the only identified group that are strongly analytic in orientation (.19). They tend to work in epistemology (.16) or philosophy of language (.12), and are rarely found in 17th-19th century (-.12) or continental (-.11) philosophy. They're influenced by notorious two-boxer and modal realist David Lewis (.1), and show an aversion to Hegel (-.12), Aristotle (-.11), and and Wittgenstein (-.1).
An observation: Different philosophers rely on — and fall victim to — substantially different groups of methods and intuitions. A few simple heuristics, like 'don't believe weird things until someone conclusively demonstrates them' and 'believe things that seem to be important metaphysical correlates for basic human institutions' and 'fall in love with any views starting with "ext"', explain a surprising amount of diversity. And there are clear common tendencies to either trust one's own rationality or to distrust it in partial (Externalism) or pathological (Anti-Realism, Anti-Naturalism) ways. But the heuristics don't hang together in a single Philosophical World-View or Way Of Doing Things, or even in two or three such world-views.
There is no large, coherent, consolidated group that's particularly attractive to LWers across the board, but philosophers seem to fall short of LW expectations for some quite distinct reasons. So attempting to criticize, persuade, shame, praise, or even speak of or address philosophers as a whole may be a bad idea. I'd expect it to be more productive to target specific 'load-bearing' doctrines on dimensions like the above than to treat the group as a monolith, for many of the same reasons we don't want to treat 'scientists' or 'mathematicians' as monoliths.
Another important result: Something is going seriously wrong with the high-level training and enculturation of professional philosophers. Or fields are just attracting thinkers who are disproportionately bad at critically assessing a number of the basic claims their field is predicated on or exists to assess.
Philosophers working in decision theory are drastically worse at Newcomb than are other philosophers, two-boxing 70.38% of the time where non-specialists two-box 59.07% of the time (normalized after getting rid of 'Other' answers). Philosophers of religion are the most likely to get questions about religion wrong — 79.13% are theists (compared to 13.22% of non-specialists), and they tend strongly toward the Anti-Naturalism dimension. Non-aestheticians think aesthetic value is objective 53.64% of the time; aestheticians think it's objective 73.88% of the time. Working in epistemology tends to make you an internalist, philosophy of science tends to make you a Humean, metaphysics a Platonist, ethics a deontologist. This isn't always the case; but it's genuinely troubling to see non-expertise emerge as a predictor of getting any important question in an academic field right.
EDIT: I've replaced "cluster" talk above with "dimension" talk. I had in mind gjm's "clusters in philosophical idea-space", not distinct groups of philosophers. gjm makes this especially clear:
I'm particularly guilty of promoting this misunderstanding (including in portions of my own brain) by not noting that the dimensions can be flipped to speak of (anti-anti-)naturalists, anti-rationalists, etc. My apologies. As Douglas_Knight notes below, "If there are clusters [of philosophers], PCA might find them, but PCA might tell you something interesting even if there are no clusters. But if there are clusters, the factors that PCA finds won't be the clusters, but the differences between them. [...] Actually, factor analysis pretty much assumes that there aren't clusters. If factor 1 put you in a cluster, that would tell pretty much all there is to say and would pin down your factor 2, but the idea in factor analysis is that your factor 2 is designed to be as free as possible, despite knowing factor 1."