Posted by Callum  10/19/2017 7:26 pm | #1 |
So I found these quite pressing points from a review of TLS against realism and in favour of nominalism regarding universals. I wonder what thoughts people have?
"In his existential detective story, Why Does the World Exist?, the American philosopher Jim Holt, explains the nominalist view of mathematics:
All of mathematics can be seen to consist of if-then propositions: if such-and-such structure satisfies certain conditions, then that structure must satisfy certain further conditions. These if-then truths are indeed logically necessary. But they do not entail the existence of any objects, whether abstract or material. The proposition "2 + 2 = 4," for example, tells you that if you had two unicorns and you added two more unicorns,then you would end up with four unicorns. But this if-then proposition is true even in a world that is devoid of unicorns—or, indeed, in a world that contains nothing at all. (181)
The fact that different minds can independently arrive at the concept that 2 + 2 = 4 does not mean that numbers exist "out there" in some "realm," it means that if you have two objects, and add two more objects, you'll have four objects. So logically, 2 + 2 = 4 and it cannot be any other way. For additional arguments against the realist view on numbers and a positive argument for nominalism, see Roberto Unger and Lee Smolin's new book, The Singular Universe and the Reality of Time: A Proposal In Natural Philosophy. In it, Smolin devotes a chapter to arguing that mathematics is indeed a human invention, but a particular kind. It's not an arbitrary invention, but rather one thatevokes objective things, analogous (albeit imperfectly) to the way chess has rules that allow anyone to objectively discover which moves are the best for which situations. Math as they see it, works the same way.
When a game like chess is invented a whole bundle of facts become demonstrable, some of which indeed are theorems that become provable through straightforward mathematical reasoning. As we do not believe in timeless Platonic realities, we do not want to say that chess always existed — in our view of the world, chess came into existence at the moment the rules were codified. This means we have to say that all the facts about it became not only demonstrable, but true, at that moment as well … Once evoked , the facts about chess are objective, in that if any one person can demonstrate one, anyone can. And they are independent of time or particular context: they will be the same facts no matter who considers them or when they are considered.
[...]
There is a potential infinity of formal axiomatic systems (FASs). Once one is evoked it can be explored and there are many discoveries to be made about it. But that statement does not imply that it, or all the infinite number of possible formal axiomatic systems, existed before they were evoked. Indeed, it’s hard to think what belief in the prior existence of an FAS would add. Once evoked, an FAS has many properties which can be proved about which there is no choice — that itself is a property that can be established. This implies there are many discoveries to be made about it. In fact, many FASs once evoked imply a countably infinite number of true properties, which can be proved.” (423-425)
In other words, the apparent dichotomy that numbers are either invented (nominalism) or discovered (realism) is a false dichotomy. Smolin outlines four possibilities. There's also no reason why these formal axiomatic systems cannot apply just as easily to geometric Forms as they do to numbers. (For additional information on Smolin's argument, seeMassimo Pigliucci's review of it here.)".
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Posted by Callum 10/19/2017 7:26 pm | #2 |
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Posted by seigneur 10/20/2017 1:50 am | #3 |
I looked at Pigliucci's review. The table with the four slots does not seem to do away with the dichotomy. Instead, it operates on the same. When you propose that numbers (or mathematical objects/axioms) either existed prior or did not exist prior, then how is it different from saying that they are real or not? Since there are four options, instead of two, Smolin has simply added another dimension to the existing dichotomy.
Smolin's overall positive argument is not too different from other nominalism I have seen: Look back at history, see how people discover physical objects at first, then they formalize the discovery, and then they keep elaborating on the formalization. Well, nobody denies the story, but there's a problem with the hidden premises that the objects are real while the formalization is a mere abstraction all the way. In reality, the formalization could be a representation of actual laws of nature which make the physical objects behave the way they do, permitting us to verify and improve on whatever formalizations we come up with.
(My glance was quick. I might have missed something. Maybe there was something special after all.)
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Posted by Callum 10/20/2017 6:04 am | #4 |
Thanks for the reply.
The difficulty I have is with the chess analogy. Its seems to account for the necessity and objectivity in maths that I thought nominalism couldn't achieve.
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Posted by Callum 10/20/2017 6:09 am | #5 |
seigneur wrote:
there's a problem with the hidden premises that the objects are real while the formalization is a mere abstraction all the way. In reality, the formalization could be a representation of actual laws of nature which make the physical objects behave the way they do, permitting us to verify and improve on whatever formalizations we come up with.
(My glance was quick. I might have missed something. Maybe there was something special after all.)
So is this the idea that the axioms or formalisations are not mind-dependent and that Smolin hasn't done enough to justify mind-dependence over independence?
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Posted by Callum 10/20/2017 6:10 am | #6 |
Do you have any books on the topic which you found helpful on the realism/nominalism debate?
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Posted by seigneur 10/20/2017 7:14 am | #7 |
Thanks for the reply.
The difficulty I have is with the chess analogy. Its seems to account for the necessity and objectivity in maths that I thought nominalism couldn't achieve.
How? The rules of chess of course apply to chess, but do they apply to anything other than chess? Yes, board games can be evoked as described: we invent the game, build its pieces, set the rules and then we can play. The real world may look kind of analogous, but the procedure is reversed: whether we know the rules (laws of nature) or not, they are already pre-applied to us and we have to follow them regardless whether we like it or not. If we want to be able to follow the rules easier, we have to learn more about them (i.e. discover rather than invent). In terms of the chess analogy, we are the pieces in the game, the rules that apply to us have been evoked by someone else, and our role is to observe and obey the moves. We cannot make any innovations to what really matters, such as laws of nature. We are on an amazingly long leash in terms of the ability to make all sorts of choices, but the leash is clearly there.
You can't point your fist towards the moon and fly there like Superman. The nominalist may answer that he is not so silly as to want to do that, that he is a rational human being. However, movies about Superman show that people with vivid imagination want to do that, while in reality we can't, so it's a valid example of how we have to correct our wishful thinking vis-a-vis reality. The philosophy that affirms that such correction is operative in the universe and vital for us to check our sanity is called realism.
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Posted by Calhoun 10/23/2017 12:56 pm | #8 |
"In his existential detective story, Why Does the World Exist?, the American philosopher Jim Holt, explains the nominalist view of mathematics:
All of mathematics can be seen to consist of if-then propositions: if such-and-such structure satisfies certain conditions, then that structure must satisfy certain further conditions. These if-then truths are indeed logically necessary. But they do not entail the existence of any objects, whether abstract or material. The proposition "2 + 2 = 4," for example, tells you that if you had two unicorns and you added two more unicorns,then you would end up with four unicorns. But this if-then proposition is true even in a world that is devoid of unicorns—or, indeed, in a world that contains nothing at all. (181)
I haven't read that book but looking at this passage I think there must be more to it than you quote, because this doesn't even remotely sound like an argument for nominalism, absent further argumentation all this seems to suggest is that numbers can be reduced to propositions, but how does that lead to nominalism? Realism isn't just realism about numbers. it also includes possible worlds, universals, propositions etc..
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Posted by Callum 10/25/2017 1:30 pm | #9 |
Thanks for the reply.
The difficulty I have is with the chess analogy. Its seems to account for the necessity and objectivity in maths that I thought nominalism couldn't achieve.How? The rules of chess of course apply to chess, but do they apply to anything other than chess? Yes, board games can be evoked as described: we invent the game, build its pieces, set the rules and then we can play. The real world may look kind of analogous, but the procedure is reversed: whether we know the rules (laws of nature) or not, they are already pre-applied to us and we have to follow them regardless whether we like it or not. If we want to be able to follow the rules easier, we have to learn more about them (i.e. discover rather than invent). In terms of the chess analogy, we are the pieces in the game, the rules that apply to us have been evoked by someone else, and our role is to observe and obey the moves. We cannot make any innovations to what really matters, such as laws of nature. We are on an amazingly long leash in terms of the ability to make all sorts of choices, but the leash is clearly there.
You can't point your fist towards the moon and fly there like Superman. The nominalist may answer that he is not so silly as to want to do that, that he is a rational human being. However, movies about Superman show that people with vivid imagination want to do that, while in reality we can't, so it's a valid example of how we have to correct our wishful thinking vis-a-vis reality. The philosophy that affirms that such correction is operative in the universe and vital for us to check our sanity is called realism.
I think the chess analogy crucially rests on the argument that there are many formal axiomatic systems. In that sense they already exist?
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Posted by Callum 10/25/2017 1:31 pm | #10 |
"In his existential detective story, Why Does the World Exist?, the American philosopher Jim Holt, explains the nominalist view of mathematics:
All of mathematics can be seen to consist of if-then propositions: if such-and-such structure satisfies certain conditions, then that structure must satisfy certain further conditions. These if-then truths are indeed logically necessary. But they do not entail the existence of any objects, whether abstract or material. The proposition "2 + 2 = 4," for example, tells you that if you had two unicorns and you added two more unicorns,then you would end up with four unicorns. But this if-then proposition is true even in a world that is devoid of unicorns—or, indeed, in a world that contains nothing at all. (181)I haven't read that book but looking at this passage I think there must be more to it than you quote, because this doesn't even remotely sound like an argument for nominalism, absent further argumentation all this seems to suggest is that numbers can be reduced to propositions, but how does that lead to nominalism? Realism isn't just realism about numbers. it also includes possible worlds, universals, propositions etc..
Right, that was the weak part I thought. It was Unger and Smolin's part I was mainly concerned with.
