Consider a squirrel on a tree. There are things we can represent that are simply not within the capacity of the squirrel's mental structure to represent. For instance, we can represent complex mathematical truths about the shapes and physical relations obtaining in the squirrel's environment. Despite the fact that all these facts are "happening" right in his face, our little squirrel hasn't the slightest clue about them. And it's not just that he's ignorant about them like someone who doesn't know math or physics; it's that it's completely beyond a mind like his to even represent things like facts and propositions. They are completely ineffable to the squirrel.
That all seems fair enough. But here's where things start to become worrisome: Why shouldn't we think that we are in relation to some possible being in the same way that the squirrel is related to us? (Or, if you believe in God, why not think that we are related to some actual being in the same way that the squirrel is related to us?) In other words, why not think that there are some aspects of reality that are completely ineffable to us, even if maybe they are effable with respect to some other, more advanced being? Why not think that there could be some species of creature who could comprehend things that are completely beyond our mind's capacity to even represent?
This is not a new argument or anything. See for instance Ch. VI of Thomas Nagel's book 'The View From Nowhere'. Moreover, I'm sure many of us have thought of this possibility before. But do we really consider the implications of this argument? I know that I've thought of this before, but haven't drawn anything of significance from this. However, I'm realizing now that this is a very important and deep question.
Note that this view seems to go hand in hand with the idea of metaphysical realism, which holds that there is an objectively existing world that is in some sense totally (or at least mostly) mind-independent. If you hold that view, it seems rather strange to think that somehow all of reality, of necessity, must be in principle comprehensible to us. And if reality were in its totality to be comprehensible to us it would be a rather strange coincidence. Moreover, there seems to be nothing particularly special about us. We are on a continumm with non-sentient creatures, insects, and squirrels on one side and with angels and God on the other (and probably a lot of things in between). Hence, it seems quite likely, on realism, that we are in a situation similar to that of the squirrel: There are aspects of reality which are simply beyond the representational capacities of our minds.
But then there is trouble. Interestingly, those who have high hopes for metaphysics tend to be metaphysical realists, but that very same metaphysical realism tends to undermine the high hopes for metaphysics. For example, suppose we characterize metaphysics as the study of the most fundamental or general aspects of reality. Suppose moreover that we are metaphysical realists. Then, probably, there are aspects of reality which are simply beyond the representational capacities of our minds. But in that case, for all we know, the most fundamental or general aspects of reality are within the sphere of things that are completely ineffable to us. So there is reason to doubt the possibility of having any substantial metaphysical knowledge.
In fact, maybe by a similar though distinct argument we can get a stronger conclusion. We have some reason to think that as minds become more advanced on the "great chain of being" that I've described, they become able to represent more (and more) fundamental aspects of reality than those before them. Higher beings have concepts that are more fundamental than those of the minds on lower levels. (Technically, they have concepts of things that are more fundamental.) And they probably have more of them. For instance, some lower animals can probably represent things like 'cause' and 'object' and even 'agent', but it seems doubtful whether bees could do the same (or to the same degree). But in that case, granted we are probably pretty far from the high end of the continuum, probably the most fundamental aspects of reality are only representable by beings on the higher end. So, probably, we cannot represent the most fundamental aspects of reality. So, probably, ambitious metaphysics is hopeless.
This might seem like a fun philosophical puzzle, but actually it is rather important, because if I sit down and ask myself whether I really think metaphysical realism is true, I am with utter and literal sincerity inclined to say, "Yes." And if I sit down and ask myself whether I really think some aspects of reality are ineffable for the reasons described, I am with utter and literal sincerity inclined to say, "Yes." And, to bring the trilemma to completion, I have high hopes for metaphysics and sincerely think it is essential to truly understanding the world.
What to do then? Does the argument against substantial metaphysics work? What are the implications for metaphysics and other areas of philosophy depending on which way one goes? How might different views solve the issues here? These are interesting questions. Since I've been thinking about this stuff for a class I'm taking, I'll probably have a chance to write a term paper on it. I have inklings about where we might go, but I have no clear answer at the moment.
Friday, October 30, 2015
Friday, October 16, 2015
Link: Amazing Philosophy Interviews with Bryan Magee
I just wanted to share a very special link with any readers who happen upon my blog.
Several years ago (probably about five or so years ago) I worked as a mascot for Liberty Tax Service in southern California. At the same time I was going to community college, and I had decided by this point that I wanted to study philosophy. I had only begun to learn about analytic philosophy. While waving a sign dressed up in a statue of liberty costume can be fun at first, after several hours one might feel a need for intellectual stimulation. Hence, I would download philosophy talks from several different sources, and one of them was here.
There are many full-length interviews on this Youtube channel, all available for free. They each consist of a dialogue between Bryan Magee and one of the greatest philosophers of the twentieth century. They are very interesting, and are especially helpful in the case that one is not already familiar with the concepts being discussed (though even then they are still fun to listen to). If you're not familiar with some area of philosophy they can give you a quick crash-course on the topic. I really recommend checking them out. Here is one I remember enjoying a lot, with A.J. Ayer talking about logical positivism. One of the best lines is when Magee asks, "What do you think was the biggest issue with logical positivism?" and Ayer replies, "Well, it was all false." (Part 4, 6:26)
Several years ago (probably about five or so years ago) I worked as a mascot for Liberty Tax Service in southern California. At the same time I was going to community college, and I had decided by this point that I wanted to study philosophy. I had only begun to learn about analytic philosophy. While waving a sign dressed up in a statue of liberty costume can be fun at first, after several hours one might feel a need for intellectual stimulation. Hence, I would download philosophy talks from several different sources, and one of them was here.
There are many full-length interviews on this Youtube channel, all available for free. They each consist of a dialogue between Bryan Magee and one of the greatest philosophers of the twentieth century. They are very interesting, and are especially helpful in the case that one is not already familiar with the concepts being discussed (though even then they are still fun to listen to). If you're not familiar with some area of philosophy they can give you a quick crash-course on the topic. I really recommend checking them out. Here is one I remember enjoying a lot, with A.J. Ayer talking about logical positivism. One of the best lines is when Magee asks, "What do you think was the biggest issue with logical positivism?" and Ayer replies, "Well, it was all false." (Part 4, 6:26)
Friday, October 9, 2015
Greg Cavin on Bayes' Theorem and Miracles
I wrote most of this post several months ago when my friend Calum Miller came to southern California for a semester abroad. Unfortunately, I simply never got around to finishing it up. Hence, this post comes about five or six months late. However, I still think it's worth posting, in case someone watches the video or comes upon the type of fallacy that I suspect goes into the argument. Here's the post:
A couple weeks ago I went to a debate between my friend Calum Miller and philosopher Greg Cavin on the Resurrection of Jesus. The video can be found here. Cavin's opening speech on Bayes starts at (6:00). He gets into his arguments again at (13:40). In this post I'll discuss a small part of Cavin's opening speech.
At the beginning, Cavin claims that he will show that it is "virtually 100% certain that no miracles ever occur."
Greg Cavin formulates the argument in terms of an assessment of a comparison between probabilities. While Cavin goes into a ton of mathematical detail that I suspect could be simplified to get to the main point, a little bit of it is probably necessary. He formulates the argument in terms of the Odds Form of Bayes' Theorem.
In general, the Odds Form of Bayes' Theorem is as follows. For any events A, B, and D:
P(A|D)/P(B|D) = P(D|A)/P(D|B) * P(A)/P(B)
Cavin comes up with a partition of probability space which is exhaustive and exclusive. In other words, at least one of the following hypotheses holds and if one holds then the others do not.
M and L are taken to be incompatible because if a miracle occurs, i.e. if M is true, then that entails the failure of at least one "un-provisoed" law of science at some time and place, whereas L entails all "un-provisoed" laws of science hold at all times and places. Cavin defines the evidence E with respect to which we will evaluate these probabilites as follows (27:00):
.jpg)
Applying Bayes' Theorem to the argument at hand, this is the Ratio of Posterior Probabilities of L vs M:
P(L|E)/P(M|E) = P(E|L)/P(E|M) * P(L)/P(M)
In other words, the left hand side compares the likelihood of L given the evidence with the likelihood of M given the evidence.
Now, a crucial part of Cavin's argument is in calculating the ratio P(E|L)/P(E|M). This is done by calculating probabilities of all of the Ti given L and M and calculating the probabilities of all the Ci given L and M. If these are lower on M than they are on L, then P(E|L) will be higher than P(E|M). His official argument here is from (31:00) to (34:00), but I asked him a question later that gets to the same point.
After the talk, I asked Cavin why he thought M could not explain C and T as well as L could. In other words, why are, say, the confirmation instances of science less likely given that miracles have occurred than if L holds? He said, "Well, if I told you, 'This is a desk,' what would that explain? Not much. How can you make any predictions from that? So, likewise, how can the proposition that at least one miracle holds explain anything? It could hardly have any predictive or explanatory power." Of course, that seems true. If the only sentence you knew to be true were "At least one miracle occurs," then you wouldn't be able to predict much, just as you couldn't predict much from just knowing "This is a desk." Hence, the argument goes, P(E|M) is very low.
However, it's a little bit misleading to put things this way. P(E|M), strictly speaking, isn't defined in terms of how much you can predict from the single proposition that at least one miracle occurs. This is clear after considering some very basic probability theory.
First, note that we can always define P(M) as P(M∩A) + P(M∩¬A) for any event A. This can clearly be seen by the following diagram:
Suppose that 'A' denotes some hypothesis, maybe the hypothesis 'The laws of nature almost always but not always hold.' Then the probability that some miracle happens is equal to the probability that some miracle happens and A holds plus the probability that some miracle happens and A does not hold. Again, P(M) = P(M∩A) + P(M∩¬A).
From this we can infer: P(E|M) = P(E|M∩A) + P(E|M∩¬A). Now, you might still think that this is lower than P(E|L) for various reasons. But you certainly couldn't infer it from the type of argument I sketched above. That would be much too easy.
Maybe I am misrepresenting what Cavin said. I hope I'm not. But if I am, let's just say that if someone were to argue in the way I represented Cavin as arguing, then they would be committing a fallacy.
There were many other interesting issues that came up during the debate, such as the likelihood of the laws of nature holding most of the time given theism, and these deserve attention. But for now I think it's worth noting that Cavin's argument doesn't go through as easily as it might have seemed.
A couple weeks ago I went to a debate between my friend Calum Miller and philosopher Greg Cavin on the Resurrection of Jesus. The video can be found here. Cavin's opening speech on Bayes starts at (6:00). He gets into his arguments again at (13:40). In this post I'll discuss a small part of Cavin's opening speech.
At the beginning, Cavin claims that he will show that it is "virtually 100% certain that no miracles ever occur."
Greg Cavin formulates the argument in terms of an assessment of a comparison between probabilities. While Cavin goes into a ton of mathematical detail that I suspect could be simplified to get to the main point, a little bit of it is probably necessary. He formulates the argument in terms of the Odds Form of Bayes' Theorem.
In general, the Odds Form of Bayes' Theorem is as follows. For any events A, B, and D:
P(A|D)/P(B|D) = P(D|A)/P(D|B) * P(A)/P(B)
Cavin comes up with a partition of probability space which is exhaustive and exclusive. In other words, at least one of the following hypotheses holds and if one holds then the others do not.
- M: At least one miracle has, had, or will occur in the universe.
- L: The laws of the sciences as these are currently formulated in standard reference works, without any supernatural non-interference proviso, are true and are laws of nature in their restricted domains.
- (¬M & ¬L): Neither M nor L hold.
M and L are taken to be incompatible because if a miracle occurs, i.e. if M is true, then that entails the failure of at least one "un-provisoed" law of science at some time and place, whereas L entails all "un-provisoed" laws of science hold at all times and places. Cavin defines the evidence E with respect to which we will evaluate these probabilites as follows (27:00):
- E: The total evidence, which is a combination of T & C, where T and C are understood as follows:
- T: All of the traces (call them Ti) of miracles. These are all of the pieces of evidence people could take to provide evidence for a miracle.
- C: All of the confirmation instances (call them Ci) of the laws of science. These are all of the pieces of evidence people could take to provide evidence for the various scientific laws.
.jpg)
The partitioning of probability space.
P(L|E)/P(M|E) = P(E|L)/P(E|M) * P(L)/P(M)
In other words, the left hand side compares the likelihood of L given the evidence with the likelihood of M given the evidence.
Now, a crucial part of Cavin's argument is in calculating the ratio P(E|L)/P(E|M). This is done by calculating probabilities of all of the Ti given L and M and calculating the probabilities of all the Ci given L and M. If these are lower on M than they are on L, then P(E|L) will be higher than P(E|M). His official argument here is from (31:00) to (34:00), but I asked him a question later that gets to the same point.
After the talk, I asked Cavin why he thought M could not explain C and T as well as L could. In other words, why are, say, the confirmation instances of science less likely given that miracles have occurred than if L holds? He said, "Well, if I told you, 'This is a desk,' what would that explain? Not much. How can you make any predictions from that? So, likewise, how can the proposition that at least one miracle holds explain anything? It could hardly have any predictive or explanatory power." Of course, that seems true. If the only sentence you knew to be true were "At least one miracle occurs," then you wouldn't be able to predict much, just as you couldn't predict much from just knowing "This is a desk." Hence, the argument goes, P(E|M) is very low.
However, it's a little bit misleading to put things this way. P(E|M), strictly speaking, isn't defined in terms of how much you can predict from the single proposition that at least one miracle occurs. This is clear after considering some very basic probability theory.
First, note that we can always define P(M) as P(M∩A) + P(M∩¬A) for any event A. This can clearly be seen by the following diagram:
P(M) = P(M∩A) + P(M∩¬A)
A is marked out in dark blue.
¬A is marked out in light blue.
Suppose that 'A' denotes some hypothesis, maybe the hypothesis 'The laws of nature almost always but not always hold.' Then the probability that some miracle happens is equal to the probability that some miracle happens and A holds plus the probability that some miracle happens and A does not hold. Again, P(M) = P(M∩A) + P(M∩¬A).
From this we can infer: P(E|M) = P(E|M∩A) + P(E|M∩¬A). Now, you might still think that this is lower than P(E|L) for various reasons. But you certainly couldn't infer it from the type of argument I sketched above. That would be much too easy.
Maybe I am misrepresenting what Cavin said. I hope I'm not. But if I am, let's just say that if someone were to argue in the way I represented Cavin as arguing, then they would be committing a fallacy.
There were many other interesting issues that came up during the debate, such as the likelihood of the laws of nature holding most of the time given theism, and these deserve attention. But for now I think it's worth noting that Cavin's argument doesn't go through as easily as it might have seemed.
Tuesday, October 6, 2015
Shameless Hyperintensionalism in Ethics (And Other Areas)
A hyperintensional position in a sentence is one where substitution of necessarily co-extensional statements does not preserve truth value. So for instance, 'believes' is a hyperintensional position. Alfredo believes triangles all have three sides doesn't necessarily imply Alfredo believes that triangles all have angles adding up to 180 degrees (and vice versa). After all, I might not know this yet.
What I will call a metaphysical hyperintensional position (and which I will just call 'hyperintensional' hereafter) is, intuitively, one where the hyperintensionality doesn't arise because of some mental attitude. This can be defined more precisely, but for my purposes some examples will suffice.
For instance, the operator 'essentially' is hyperintensional, at least on one understanding of 'essentially.' To use a historical example, Socrates is essentially a rational animal; he is not essentially risible, though necessarily if something has the one property it has the other. Or to use the more contemporary example, Socrates is not essentially a member of his singleton set -- this doesn't have to do with what he is, at the most fundamental level, in himself -- even though he has the property of being so necessarily. Grounding, intrinsicality, naturalness, reduction; these all seem to be hyperintensional as well.
Many metaphysicians are skeptical of these concepts. (Though of course many are not! Which is why they are such a hot topic of discussion lately.) I think there is usually some ambiguity in what this "skepticism" amounts to, but it is often made explicit in terms of the good old, "I don't know what that means." This skepticism looms especially large among those of a certain breed of metaphysician, whose generation either made advances over extensional concepts by employing intensional ones, or learned from those who did. (On all of this, see Daniel Nolan's very enjoyable paper.)
One thing I find worth noting is that people in ethics use these concepts shamelessly, both in first-order normative ethics and in meta-ethics. Meta-ethical claims are consistently stated in terms of "in virtue of" (just witness the Euthyphro Dilemma) or "grounding." There is a consistent flow of talk about what is essential to an action (as opposed to what is just necessarily true of it), as well as the intrinsic features of an action. Ethicists seek real definitions of what's right and wrong, seek to categorize things into natural kinds, seek to reduce properties to other ones, and seek to explain less fundamental facts in terms of more fundamental principles; in general, ethicists have no qualms about using metaphysical hyperintensional concepts, while many metaphysicians claim skepticism even about their intelligibility.
Philosophers of mind initially seem to be a bit more careful in this regard, at least those working in the metaphysics of mind. Witness the debates about supervenience for instance. However, it seems many philosophers of mind are realizing that they have really been trying to raise issues that can only be adequately stated using hyperintensional language. And many times even those who are keen to phrase things in terms of supervenience will explicitly distinguish this purely modal notion from what's really doing the work (explanation, grounding, reduction, and so forth).
Many perfectly legitimate metaphysical debates themselves are best phrased in terms of hyperintensional concepts. And if we look in areas other than metaphysics, a very similar pattern seems to arise; think of philosophy of science, philosophy of math, philosophy of religion, free will and moral responsibility, etc.
One of the clearest examples though is in ethics. It might be worth it for me to demonstrate my empirical claims with specific examples (though frankly it's harder for me to think of papers in ethics which don't make any use of these concepts than those which do). I might try to do this at some point. But I think it's just worth pointing out for now: The sense I get from everything I've read in ethics over the years is that ethicists have no problem whatsoever using any of the hyperintensional notions that metaphysicians will sometimes claim to have no understanding of. Obviously that is not universally true, and some positions in ethics are in fact more "deflationary" than others. But arguably this is not the norm.
Considering that ethics is often one of the closest areas of philosophy to "real world" problems this is important, since it lends some weight toward thinking that these are not arcane, idiosyncratic notions, but would be considered completely intelligible to most people (and they are; anyone taking intro to ethics will "get" the Euthyphro Dilemma; in fact, their understanding of this problem will probably be much clearer than any problem involving modality for example). Also, if you grant a certain level of epistemic autonomy to ethics -- it doesn't need to wait on the approval of metaphysicians to be considered legitimate -- then it would be wrong to pronounce, based on an ill-defined "skepticism," that these concepts are unintelligible. The interaction between metaphysics and ethics should really be one of reflective equilibrium. And in that case, the initial reaction to hyperintensional concepts should be one of cautious approval rather than default skepticism.
What I will call a metaphysical hyperintensional position (and which I will just call 'hyperintensional' hereafter) is, intuitively, one where the hyperintensionality doesn't arise because of some mental attitude. This can be defined more precisely, but for my purposes some examples will suffice.
For instance, the operator 'essentially' is hyperintensional, at least on one understanding of 'essentially.' To use a historical example, Socrates is essentially a rational animal; he is not essentially risible, though necessarily if something has the one property it has the other. Or to use the more contemporary example, Socrates is not essentially a member of his singleton set -- this doesn't have to do with what he is, at the most fundamental level, in himself -- even though he has the property of being so necessarily. Grounding, intrinsicality, naturalness, reduction; these all seem to be hyperintensional as well.
Many metaphysicians are skeptical of these concepts. (Though of course many are not! Which is why they are such a hot topic of discussion lately.) I think there is usually some ambiguity in what this "skepticism" amounts to, but it is often made explicit in terms of the good old, "I don't know what that means." This skepticism looms especially large among those of a certain breed of metaphysician, whose generation either made advances over extensional concepts by employing intensional ones, or learned from those who did. (On all of this, see Daniel Nolan's very enjoyable paper.)
One thing I find worth noting is that people in ethics use these concepts shamelessly, both in first-order normative ethics and in meta-ethics. Meta-ethical claims are consistently stated in terms of "in virtue of" (just witness the Euthyphro Dilemma) or "grounding." There is a consistent flow of talk about what is essential to an action (as opposed to what is just necessarily true of it), as well as the intrinsic features of an action. Ethicists seek real definitions of what's right and wrong, seek to categorize things into natural kinds, seek to reduce properties to other ones, and seek to explain less fundamental facts in terms of more fundamental principles; in general, ethicists have no qualms about using metaphysical hyperintensional concepts, while many metaphysicians claim skepticism even about their intelligibility.
Philosophers of mind initially seem to be a bit more careful in this regard, at least those working in the metaphysics of mind. Witness the debates about supervenience for instance. However, it seems many philosophers of mind are realizing that they have really been trying to raise issues that can only be adequately stated using hyperintensional language. And many times even those who are keen to phrase things in terms of supervenience will explicitly distinguish this purely modal notion from what's really doing the work (explanation, grounding, reduction, and so forth).
Many perfectly legitimate metaphysical debates themselves are best phrased in terms of hyperintensional concepts. And if we look in areas other than metaphysics, a very similar pattern seems to arise; think of philosophy of science, philosophy of math, philosophy of religion, free will and moral responsibility, etc.
One of the clearest examples though is in ethics. It might be worth it for me to demonstrate my empirical claims with specific examples (though frankly it's harder for me to think of papers in ethics which don't make any use of these concepts than those which do). I might try to do this at some point. But I think it's just worth pointing out for now: The sense I get from everything I've read in ethics over the years is that ethicists have no problem whatsoever using any of the hyperintensional notions that metaphysicians will sometimes claim to have no understanding of. Obviously that is not universally true, and some positions in ethics are in fact more "deflationary" than others. But arguably this is not the norm.
Considering that ethics is often one of the closest areas of philosophy to "real world" problems this is important, since it lends some weight toward thinking that these are not arcane, idiosyncratic notions, but would be considered completely intelligible to most people (and they are; anyone taking intro to ethics will "get" the Euthyphro Dilemma; in fact, their understanding of this problem will probably be much clearer than any problem involving modality for example). Also, if you grant a certain level of epistemic autonomy to ethics -- it doesn't need to wait on the approval of metaphysicians to be considered legitimate -- then it would be wrong to pronounce, based on an ill-defined "skepticism," that these concepts are unintelligible. The interaction between metaphysics and ethics should really be one of reflective equilibrium. And in that case, the initial reaction to hyperintensional concepts should be one of cautious approval rather than default skepticism.
Friday, October 2, 2015
Review: 'An Aristotelian Realist Philosophy of Mathematics' by James Franklin
I recently finished reading James Franklin's marvelous book, An Aristotelian Realist Philosophy of Mathematics, and I want to advertise it here. This is a great book. It is empirically informed by a wide knowledge of both actual mathematical practice and contemporary mathematics itself, along with other relevant areas of study such as perceptual psychology, neuroscience, and engineering. It also engages with much of the cutting edge in contemporary philosophy of mathematics, especially in the later chapters. This is some of the best of what Aristotelianism has to offer. I really hope people will read it.
Franklin aims to give an account of mathematics as the science of quantity and of structure. Franklin gives particularly clear definitions of both quantity and structure--something often lacking among contemporary structuralists in my opinion--and this in itself is a very important advance. According to his account, mathematics studies structural universals and quantities. These universals and quantities are the type of thing that can be found in the real world and can be literally had by concrete objects. Of course, not all mathematical structures are had by some concrete object, but it is essential to his account that they could be, i.e. that they are metaphysically possible.
While quantity seems to me to play a less central part in his project, his clear account of structure allows him to take his views a long way. Franklin understands a property to be purely structural just in case it can be defined completely in terms of 'part', 'whole', 'same', 'different', and purely logical vocabulary. The relations of 'part' and 'whole' will probably come into play in geometry, as well as set theory, graph theory, topology, analysis, etc. So, for instance, on this definition, the property of being a Euclidean space could probably be defined purely structurally; see for instance Hilbert's axioms. Also, the Peano axioms seem to describe purely structural relations, since they only invoke logical vocabulary and identity (other than the names for the relations being defined, of course). Franklin gives many more examples, so I refer the reader to his book for a treatment of further cases.
Franklin contrasts his approach with Platonism and nominalism in contemporary philosophy of mathematics. Unlike Platonism, the universals studied by mathematics can be literally instantiated by concrete things in the real world. What mathematics does is study these possibly instantiated structures. Mathematics does not study abstract, particular individuals. Number systems, for instance, would not be cashed out as consisting of abstract individuals (numbers), but as either systems of quantities or as structures which can be instantiated by concrete things. (Franklin's account of number, in fact, cashes out numbers as being relations which are literally instantiated in the world by material heaps and 'unit-making' universals.)
Against nominalism on the other hand, Franklin assumes that there are, in fact, mathematical universals that can be literally shared by different things. Again, Franklin also assumes that there are, in addition to those universals instantiated in the real world, universals which are not instantiated but are at least possibly instantiated.
By his choice of example he shows how contemporary philosophers of mathematics often miss the most central cases of mathematics. Contemporary philosophy of mathematics often has a Platonist bias, focusing on those cases that are less essential for use in real world applications (such as huge sets, large infinities, etc.). This is to the detriment of the most central and basic cases, which are the simple, often discrete and finite structures widely used in real-world applied sciences, and which are less amenable to Platonist interpretation.
He gives a far more plausible account of mathematical knowledge and empirical mathematical application than that offered by most Platonists. He also argues that contemporary philosophy of mathematics tends to not pay enough to attention to how mathematics is actually done, and therefore misses those aspects of mathematical practice that make more sense on an Aristotelian view. He shows a much closer parallel between actual mathematical practice and actual empirical scientific practice than is often recognized (for instance, by the unquestionable use of induction, plausible reasoning, and explanation in mathematics; he rightly notes that (in)formal proof is often only the last step in the equation). Franklin goes on to apply the Aristotelian conception of mathematics to many other philosophical issues, such as mathematical necessity, infinity, approximation, and ontology.
With that said, there are several parts of the theory that could be potentially problematic and call for more investigation. Just to shotgun a few of them out:
One last potential criticism that I feel kind of bad about making: I feel like the book doesn't really engage much with what's been said in contemporary neo-Aristotelian metaphysics and ontology. I feel bad about saying that because of the huge swaths of literature the book does, in fact, engage with (the number of works referenced is amazing; one wonders how somebody can read so much). But in certain respects (the mereology for instance, or the role of states of affairs), it seems like the book draws on some concepts with which many current Aristotelians might take issue. And like I said, the book's understanding of ontological commitment could have been a bit more clear; here, engagement with contemporary Aristotelian metaphysics (among others) might have been helpful as well.
Overall though this is an excellent book, and maybe even a game-changer, at least for me. It contains many more interesting ideas and arguments to grapple with than I've been able to discuss here. Whether one buys into it or not, Franklin admirably demonstrates the fruits of an Aristotelian approach, at least on one understanding of that term. He makes use of a wide variety of examples, from a wide variety of real world sciences (including, but very much not limited to, pure mathematics). By doing this he demonstrates how important it is to pay attention to actual empirical results and practice when doing any sort of metaphysical or epistemological investigation into the philosophical status of mathematics. And this seems to me to be one of the most important marks of the general Aristotelian attitude.
Franklin aims to give an account of mathematics as the science of quantity and of structure. Franklin gives particularly clear definitions of both quantity and structure--something often lacking among contemporary structuralists in my opinion--and this in itself is a very important advance. According to his account, mathematics studies structural universals and quantities. These universals and quantities are the type of thing that can be found in the real world and can be literally had by concrete objects. Of course, not all mathematical structures are had by some concrete object, but it is essential to his account that they could be, i.e. that they are metaphysically possible.
While quantity seems to me to play a less central part in his project, his clear account of structure allows him to take his views a long way. Franklin understands a property to be purely structural just in case it can be defined completely in terms of 'part', 'whole', 'same', 'different', and purely logical vocabulary. The relations of 'part' and 'whole' will probably come into play in geometry, as well as set theory, graph theory, topology, analysis, etc. So, for instance, on this definition, the property of being a Euclidean space could probably be defined purely structurally; see for instance Hilbert's axioms. Also, the Peano axioms seem to describe purely structural relations, since they only invoke logical vocabulary and identity (other than the names for the relations being defined, of course). Franklin gives many more examples, so I refer the reader to his book for a treatment of further cases.
Franklin contrasts his approach with Platonism and nominalism in contemporary philosophy of mathematics. Unlike Platonism, the universals studied by mathematics can be literally instantiated by concrete things in the real world. What mathematics does is study these possibly instantiated structures. Mathematics does not study abstract, particular individuals. Number systems, for instance, would not be cashed out as consisting of abstract individuals (numbers), but as either systems of quantities or as structures which can be instantiated by concrete things. (Franklin's account of number, in fact, cashes out numbers as being relations which are literally instantiated in the world by material heaps and 'unit-making' universals.)
Against nominalism on the other hand, Franklin assumes that there are, in fact, mathematical universals that can be literally shared by different things. Again, Franklin also assumes that there are, in addition to those universals instantiated in the real world, universals which are not instantiated but are at least possibly instantiated.
By his choice of example he shows how contemporary philosophers of mathematics often miss the most central cases of mathematics. Contemporary philosophy of mathematics often has a Platonist bias, focusing on those cases that are less essential for use in real world applications (such as huge sets, large infinities, etc.). This is to the detriment of the most central and basic cases, which are the simple, often discrete and finite structures widely used in real-world applied sciences, and which are less amenable to Platonist interpretation.
He gives a far more plausible account of mathematical knowledge and empirical mathematical application than that offered by most Platonists. He also argues that contemporary philosophy of mathematics tends to not pay enough to attention to how mathematics is actually done, and therefore misses those aspects of mathematical practice that make more sense on an Aristotelian view. He shows a much closer parallel between actual mathematical practice and actual empirical scientific practice than is often recognized (for instance, by the unquestionable use of induction, plausible reasoning, and explanation in mathematics; he rightly notes that (in)formal proof is often only the last step in the equation). Franklin goes on to apply the Aristotelian conception of mathematics to many other philosophical issues, such as mathematical necessity, infinity, approximation, and ontology.
With that said, there are several parts of the theory that could be potentially problematic and call for more investigation. Just to shotgun a few of them out:
- The reliance on a classical mereology of heaps and arbitrary sums (this is important for his definitions of whole numbers and sets).
- The reliance on (immanent) universals, problematic from a trope nominalist perspective such as my own, and which might use a bit more explanation.
- The commitment to uninstantiated universals (an idea classically denied by most Aristotelians, including Aristotle himself, and one which moves Franklin's account toward a "semi-Platonism" as he calls it).
- His commitment to all mathematical structures being metaphysically possible (this is interesting to me; I bet Franklin's account could be seamlessly extended given a proper account of impossibility, impossible objects, impossible universals, and impossible worlds, and I bet this isn't essential to his view).
- Giving a general, unified semantics for mathematical language (it's less than clear from the book how this is to be done; for instance, with the complex and negative numbers, Franklin gives what appear to be examples, or maybe geometrical/economical interpretations. But what would he say are the straight up truth-conditions for, say, -2 + 3i = 2(-1 + 3/2i)? Or (-2)(-3) = 6? Or of more general laws governing number systems?).
- Showing more precisely and in individual cases how a more wide range of mathematical concepts are definable either purely structurally or quantitatively (ideally, it'd be nice if we could get to the point of giving a general paraphrase scheme or a general procedure--Franklin's account of set theory being purely structural is suggestive, so maybe we could show how any set-theoretical entity or relation could be defined structurally, and thereby show all mathematics to be interpretable structurally; either this or the last question I hope to work on for my term paper this semester).
- The apparently ad hoc fictionalist account of zero and the empty set combined with a realist account of everything else (I can see fictionalists asking why we need the realistic ontology in some cases but not others).
- Related to this last point, some unclarity/implausibility in the theory of ontology and ontological commitment at play, as well as some unclarity about the ontological status of mathematics (if it were made more clear when or why we are committed to some things but not others, and in what way, it'd probably be easier to answer questions such as the last one).
One last potential criticism that I feel kind of bad about making: I feel like the book doesn't really engage much with what's been said in contemporary neo-Aristotelian metaphysics and ontology. I feel bad about saying that because of the huge swaths of literature the book does, in fact, engage with (the number of works referenced is amazing; one wonders how somebody can read so much). But in certain respects (the mereology for instance, or the role of states of affairs), it seems like the book draws on some concepts with which many current Aristotelians might take issue. And like I said, the book's understanding of ontological commitment could have been a bit more clear; here, engagement with contemporary Aristotelian metaphysics (among others) might have been helpful as well.
Overall though this is an excellent book, and maybe even a game-changer, at least for me. It contains many more interesting ideas and arguments to grapple with than I've been able to discuss here. Whether one buys into it or not, Franklin admirably demonstrates the fruits of an Aristotelian approach, at least on one understanding of that term. He makes use of a wide variety of examples, from a wide variety of real world sciences (including, but very much not limited to, pure mathematics). By doing this he demonstrates how important it is to pay attention to actual empirical results and practice when doing any sort of metaphysical or epistemological investigation into the philosophical status of mathematics. And this seems to me to be one of the most important marks of the general Aristotelian attitude.