An article I submitted years ago has finally made it into the pages of Free Inquiry magazine (issue 30.3 of April/May 2010, pp. 50-51), "On Defining Naturalism as a Worldview," part of their ongoing 'It's Only Natural' column. It was sitting in their queue for ages. It essentially just summarizes the most important points of my more extensive blog on the subject, Defining the Supernatural.It has already provoked one reply at The Teapot Atheist. But had TPA read the blog recommended in my FI article, he would have known I already addressed the concerns he raised. I just didn't have the room to fit all that into two pages of print.
First, if (per his example) we could indeed explain how karma works without any irreducible mind or mental property having caused it to exist or to operate as it does, then karma would be natural. His counter-example is thus nothing of the kind. I gave several comparable examples myself in the original blog (e.g. the difference between a Harry Potter world and a Forbidden Planet world). Hence to claim karma could exist and be a reducibly nonmental system, and thus entirely natural, only proves my point.Indeed, as it happens, such a thing does in fact exist: social reciprocity and natural consequentialism. "He who lives by the sword, dies by the sword," is a simple statement of statistical fact: certain habitual behaviors naturally cause an elevation in associated risks. Actions have consequences. This is true even more broadly: the irrational anger and stubbornness that characterizes large segments of the populations of Israel and Palestine have the direct consequence of perpetuating their misery by causing them to repeat the very actions that cause and sustain the miserable situation they loathe so much in the first place. If that's not karma, I don't know what is. But there's nothing supernatural about it.
If, on the other hand, the universe just magically "knew" what was good and evil and inexplicably always rewarded the one and punished the other, that would be an irreducibly mental behavior and thus would be undeniably supernatural (an example of "high teleology" that I actually gave in my book Sense and Goodness without God, pp. 273-75, and there in greater detail).
Unless there was some gigantic computer and system of sensors somewhere detecting what we were doing and running the math, which was programmed by some alien race that conquered the cosmos eons ago, and this was manipulating the world through natural physics to arrange for perfect reciprocal justice on earth. But then obviously no one would be calling such a thing supernatural. TPA's example of physical karmions causing karma thus leaves out the most crucial detail: how do karmions know what actions to reward or punish? Work out a complete system, without referring to any magical "knowing" or "caring," and what you'll have will be undeniably naturalistic. Just like my example of The Force in my blog (which TPA clearly didn't read). Or the "Justice Field" in Red Dwarf season 4 episode 3 ("Justice").TPA insists that "naturalism is the converse of supernaturalism" is a sufficient definition, but as I explain in FI (and prove on my blog, as well as in the articles I cite and link to in that blog), that definition is vacuously circular. Because you still have to define "supernatural." Which necessarily requires you to define "natural." Back to square one. As I proved from an analysis of what actual naturalists say they believe (see "A Brief Ethnography of Contemporary Naturalism" in Defending Naturalism as a Worldview), everyone's intuitions on this score reduce to the definition I advance in FI.
TPA has nothing to offer in its place. Instead, he simply regresses to yet another arbitrary laundry list that he can't even explain. Why is karma "clearly supernatural"? He never says, and yet if he can't, then he has no definition. He then confusingly devolves into essentially claiming naturalism is just empiricism, which is a folly I already called out in FI. That entails (and even his own examples affirm this) that "the supernatural" merely means "that which is false," which is perverse, because many supernatural things are capable of being true, and if they are, then either naturalism includes the supernatural (if God actually exists, then he no longer belongs to things that are false, and therefore naturalism would remain true, thus refuting TPA's assertion that naturalism is the converse of supernaturalism) or naturalism is false (in which case the supernatural cannot mean "that which is false" and TPA must then explain what it does mean--the very thing he doesn't do). There is no escape from this dilemma. The horns impale him.
Likewise, TPA completely misses my point about God: if God is a disembodied mind, then he is irreducibly mental, therefore supernatural. TPA would know this if he read my blog referenced in FI, where I actually described what it would take for a god to exist on naturalism (it is in fact possible--it just wouldn't be what anyone now means by the word God).
Thus, his example of God is also not a counter-example to my argument. Nor does rejecting disembodied god-minds require us to "eliminate all references to other human minds," since those minds are reducibly nonmental, thus natural. Unlike God.TPA then makes the mistake of assuming Platonism about numbers. He evidently is unaware of Nominalism and Formalism and the Aristotelian metaphysics of mathematics that I defend in my book (even though he claims to have read it; in any event, just look up "numbers, nature of" in the index of Sense and Goodness without God; and in application to physics, see my blog Our Mathematical Universe). There is no difficulty reducing numbers to nonmental things. Even so-called Platonic naturalists admit there is no way immaterial things (like Platonic numbers) can cause anything to happen (like our being aware of them), yet that renders Platonic naturalism incoherent. If immaterial abstract objects cannot cause us to know about them, how is it that we know about them? Answer: because they aren't immaterial (see "abstraction and abstract objects" in the index of Sense and Goodness without God).
TPA commits another common error of not thinking things through when he says "I have never encountered 326,519,438.004 objects," yet he just did: in his physical, reducibly nonmental brain, "nor is there anything physical about it," yet there is: the word refers to a physical fact.
Being a fraction, it refers to a ratio of two quantities, and quantity is a fundamentally physical property: space, time, matter and energy all by definition possess it. It can refer to some actual fact (odds are, such a ratio exists between some two objects in this universe--because there are so many objects in this universe to stand in ratio to each other) or a hypothetical fact (such a ratio can exist between two reducibly nonmental objects, without requiring anything irreducibly mental, e.g. I could cut two wires right now that have that ratio between them).
By analogy, there are no unicorns, either, yet the word still refers to a (hypothetical) physical fact: unicorns exist if the physical (and thus reducibly nonmental) entity described by that word exists. And if they don't exist, the thought of them exists in the brain that thinks it, and as long as the brain doing that is reducibly nonmental, then by the law of commutation, so is the idea of a unicorn. QED. What about things no one has thought of yet, but that are logically possible? They exist only potentially. Those thoughts do not actually exist until, in fact, they actually exist. If they could exist before that, then they would have to be supernatural (because they would then have to be irreducibly mental). What does it mean to potentially exist but not actually exist? It means the physical universe is capable of producing such things in the right conditions, yet those conditions still do not require anything irreducibly mental (or if they did, then naturalism would be false).
If I have a gold ring, a gold cube potentially exists. Because I can mash it into a cube. But I require no supernatural power to do that. Nothing irreducibly mental need exist for a ring of gold to be potentially a cube of gold. Thus, potentially existing things are not irreducibly mental and thus not supernatural. Hence the same follows for a ratio like 326,519,438.004: if not referring to an actual ratio (like two actual wires, or the optical distance between two stars in ratio to some arbitrary unit of distance, etc.), then it refers to a potential ratio, and potential things neither are irreducibly mental nor require the irreducibly mental (Sense and Goodness without God, pp. 125-26).TPA is perplexed at how he can think of a number but not what it describes. But that should not be perplexing. Because a potential ratio can be a ratio of anything possessing the property of quantity. Hence, obviously when we think solely of the ratio we leave blank what it is a ratio of, we thus "abstract" the ratio from its particular instances, and the resulting impression is of a number divorced from any physical fact. But that's an illusion, if we take it as anything other than a formalism of physical computation ("x:y = 326,519,438.004:1" where x and y = "wires or sticks or pounds or persons or...[ad infinitum]," there being too many possibilities to state or imagine all at once, so we don't). This was explicitly stated and explained by Aristotle over two thousand years ago. Someone really ought to get the memo.
Even when abstracted, a word like 326,519,438.004 is meaningless unless it describes some actual or hypothetical ratio between physical quantities. The blank must be potentially fillable. Otherwise numbers would be meaningless sounds. (I discuss this mistake more broadly in Sense and Goodness without God, pp. 31-32, essential to which is the whole discussion of pp. 29-35). If, on the other hand, 326,519,438.004 "exists" independently of our physical minds or computers constructing it, and not only that, but also independently of any actual or potential physical quantities, then it would certainly be supernatural.
Because there could then be no other explanation for how or why it existed at all. Failure to face the consequences of that fact can only make naturalists look ridiculous.This debate was continued in Defining Naturalism II.
24 comments:
I would say that there are concepts in mathematics which do not correspond, even potentially, to any physical quantity or property. In this category I would include numbers like aleph-omega, or concepts like the topology of the long line; these are unphysical in a very fundamental way.
(Some might argue that these things are in fact meaningless; there is some justification for this position, though I wouldn't accept it myself.)
But I wouldn't claim that these concepts are in any way supernatural or have any "existence" other than as ideas in the minds of mathematicians or as descriptions written in the language of mathematics. They arise, as I see it, as higher-order abstractions: starting from abstractions (e.g. the natural numbers) motivated by physical objects, we construct higher layers of abstraction in order to make more general statements about how those initial abstractions behave, and then higher layers above that, and so on.
In the process of doing this, we end up being able to define and describe the properties of concepts that may have meaning only within our chosen system of abstraction and not within the physical universe.
Andrew, I'd say you're right. Once I challenged a couple of people by saying that mathematics describes more than what can exist in this universe. I gave the example of negative numbers. There's no such thing, AFAIK, as negative things. I didn't get a single successful refutation (e.g. I got: temperature, electrons, money etc.). Maybe Richard will give it a shot!
It is surprising that anyone would deny that 326,519,438.004 (or any fraction) can't represent something in reality. It's just a fraction. Everyone's had (roughly) half an apple, right?
But let's say there are x particles in the universe. Is any number greater than x useful? (I assume 'yes' but don't know how). That's an interesting one to think about.
If you don't accept negative electric charges as an instance of physical existence of negative numbers I'm not sure what you would accept - it seems clear enough to me.
Numbers larger than the number of particles in the universe are indeed useful; a famous example is the number of possible chess games (or for an even larger number, the number of possible Go games).
(I deliberately chose aleph-omega as an example, rather than any smaller transfinite number, because philosophers often deny the physical existence of aleph-null while asserting the physical existence of beth-one (= 2^(aleph-null)) which is strictly larger. Aleph-omega is large enough to be well clear of this debate; even larger examples could be used, such as (alpha = aleph(alpha)), the fixed point of the aleph function. Curiously, I was just reading in Dan Barker's book how one of his debate opponents had claimed that "zero and infinity are the two simplest concepts in mathematics"; any pure mathematician familiar with the field would have a huge laugh at that.)
Are There Any Coherent Mathematical Formalisms That Do Not Correspond Even Potentially with Anything Physical?
Andrew G. said... I would say that there are concepts in mathematics which do not correspond, even potentially, to any physical quantity or property.
Everyone who says that, isn't aware of the actual physical underpinnings of mathematical formalisms. I've studied this extensively. For example, the square root of negative one refers to a physical rotation operator, realized in actual physical systems like radio circuitry (this is explained in Nahin's highly-recommended book on the imaginary number, An Imaginary Tale). Likewise, Fermat's Last Theorem describes a geometric shape. Etc.
Andrew G. said... In this category I would include numbers like aleph-omega, or concepts like the topology of the long line; these are unphysical in a very fundamental way.
The topology of a line is by definition physical (space is reducibly nonmental). If it is a line not yet realized, then it is a potential physical shape, which requires nothing to exist except a physical thing that can potentially be shaped into it (like spacetime, again a physical, nonmental thing).
Aleph-omega, if it is indeed logically possible, must be a quantity. All quantities are physical. In this case a potential quantity, but again, potential quantities require nothing else to exist but physical things. For example, if the universe will expand indefinitely (which it at least potentially can), then there will eventually in actual fact be, at t = omega (which on Relativity Theory already exists, we just haven't gotten there yet), an infinity^infinity (or aleph^aleph) volume of spacetime. Hence aleph^aleph refers to anything of such quantity, like that potential volume of spacetime (or anything else as physically numerous).
So, too, aleph-omega (which is a somewhat more elaborately defined). Cantor demonstrated the existence of cardinality by arranging quantities in a geometric matrix and showing that physically arranging them one way, you would have an infinite quantity that nevertheless excluded known quantities, adding which simply pushed out another known quantity, demonstrating that there was a quantity bigger than a merely infinite quantity. But these are still both quantities, which are potentially realizable with physical systems (Cantor’s imagined matrix could potentially be built with chalk and coconuts, if you had unlimited time and resources, both of which are at least potentially possible). Aleph-omega simply refers to another kind of arrangement like that, in some sort of physical space (multidimensional or not, using chalk and coconuts or anything else), which is either possible (and if possible, corresponds to some potential physical system), or not (and if not, it doesn’t exist, and thus is of no relevance to this discussion).
Thus, your claim that there are "concepts in mathematics which do not correspond, even potentially, to any physical quantity or property" is false.
Are There Abstractions That, Even Potentially, Only Exist in the Mind?
Andrew G. said... But I wouldn't claim that these concepts are in any way supernatural or have any "existence" other than as ideas in the minds of mathematicians or as descriptions written in the language of mathematics.
TPA’s point is that this can’t be true of everything, e.g. he says numbers must exist without mathematicians to think them--he’s part wrong: quantities exist without mathematicians to think them; numbers are words mathematicians invented to refer to those quantities; but he is right that such referents must exist apart from minds.
Nevertheless, it is true some things might exist solely as the contents of physical minds (which things would then, by commutation, be physical, too), but that entails they already potentially exist (since physical minds can potentially realize them), so if mathematicians can ever imagine any x, x potentially exists even if no mathematician ever imagines it; and if mathematicians (i.e. their minds) are reducibly nonmental, then x is reducibly nonmental (since nothing need exist for x to potentially exist except reducibly nonmental things, such as a mathematician). And since mathematicians (if naturalism is true) are potential products of a physical universe (since anything actual must also be potential), x potentially existed even before people did. It just didn't actually exist then (otherwise, x must be supernatural, unless x is physical, or otherwise reducibly nonmental).
But we can say even more than that. For if these are meaningful ideas we are talking about (i.e. if they have any cognitive content whatever, and are not logically self-contradictory) then they must necessarily be physically realizable (i.e. they must describe something that could potentially exist apart from minds). See Sense and Goodness without God pp. 29ff.
As a simple example, the only reason the axiom of infinity is accepted in mathematics is that an extension of a set by unending iteration is conceivable; and it is only conceivable because we can imagine a physical operation carrying it out, such that nothing would logically prevent it continuing (even if something might contingently do so). In other words, we can imagine counting any physical object indefinitely. Counting is a physical operation, and any physical object can be counted. And of course, imagining is the physical operation of a physical brain (if, of course, naturalism is true). Thus, in any physical universe in which brains can exist, anything mathematicians can intelligibly imagine will refer to some potentially physical fact (like endlessly counting the same stone).
I take that as a testable hypothesis. I have tested it extensively. I have consistently verified it without fail.
Does the Supernatural Potentially Exist?
Andrew G. said... [Some concepts] arise, as I see it, as higher-order abstractions
Sure. But higher-order abstractions are entirely composed of lower-order abstractions. So in every possible universe, where there are lower-order abstractions, there are necessarily higher-order abstractions (i.e. there is no logically possible thing you could do to any such universe that would keep the lower-order abstractions yet eliminate the corresponding higher-order abstractions). Thus, if physical particulars are sufficient to realize lower-order abstractions, they are necessarily sufficient to realize higher-order abstractions. Thus, nothing irreducibly mental here, either.
starting from abstractions (e.g. the natural numbers) motivated by physical objects, we construct higher layers of abstraction in order to make more general statements about how those initial abstractions behave, and then higher layers above that, and so on.
Which is all just the same process: describing actually and potentially repeatable patterns of space-time (or of matter-energy in space-time).
In the process of doing this, we end up being able to define and describe the properties of concepts that may have meaning only within our chosen system of abstraction and not within the physical universe.
Not necessarily. But certainly sometimes.
And this raises the issue of what it even means to say the supernatural is possible.
For instance, a different example of what you have in mind is discussing what must happen in a fictional world: e.g. discussing the powers and limitations of the Force in Star Wars involves describing and discussing "the properties of concepts that may have meaning only within our chosen system of abstraction [e.g. the Starwarsverse] and not within the physical universe."
Insofar as we describe the Force in the Starwarsverse supernaturally, we must posit a universe fundamentally different from ours, leaving open the question whether our universe could ever be transformed into that one (i.e. is that universe a potential state of our universe), or whether there is any other universe separate from ours that is in such a state or could be (i.e. is there another universe that is already like that universe, or that has the potential to become such), or whether there is any state of being prior to there being any universe, which produced ours but could have produced a different one, one that was or had the potential to be like that one.
If we could prove all those possibilities to be logically impossible, we will have proved all supernatural claims logically impossible. But until then we must accept that there is some nonzero epistemic probability (however small) that the supernatural was at some time (or is in some universe) a potential outcome. That need not be a metaphysical assertion (we can merely hypothesize that the supernatural is logically impossible, and that we are simply in error when we think we are imagining otherwise), nor need a naturalist assert it that way (we don’t have to believe the supernatural is logically impossible, only that it has not been realized, so far as we have any reason to believe). It can thus be merely an epistemic assertion (“I don’t know if the supernatural is logically impossible, only that, even if it was ever a potential outcome of any actual thing, it is not an actual outcome here, or anywhere else I know”).
Pikemann Urge said... Once I challenged a couple of people by saying that mathematics describes more than what can exist in this universe. I gave the example of negative numbers. There's no such thing, AFAIK, as negative things. I didn't get a single successful refutation (e.g. I got: temperature, electrons, money etc.). Maybe Richard will give it a shot!
Of course.
Look at your checkbook.
What are those negative numbers?
Physical withdrawals of actual cash (or patterns of electrons representing said cash :-).
Thus negative numbers describe a physically real fact of a purely physical universe (in this case: physical loss).
Negative numbers are more abstractable than that, since they can refer to any inversion of quantity (depth below sea level, an opposing atomic charge, financial debt, etc.).
In electronic systems, squaring a quantity (multiplying one quantity by itself) and getting -1 (in other words, imaginary numbers) corresponds to rotation out of the plane, so that a field that is aligned left-right will have quantities in each direction from center, one of which is “negative” with respect to the other, and imaginary numbers begin a third field line geometrically perpendicular to that one (so instead of numbers running from left to right of zero, we also have a number line running to the north of zero as well, and this is confirmed in observed physical phenomena like electromagnetic fields). See Nahin’s book on imaginary numbers, cited in comments earlier.
But let's say there are x particles in the universe. Is any number greater than x useful? (I assume 'yes' but don't know how).
The number of ways those particles can be connected is far greater than x and very much a useful number (at least as much as x would ever be). Andrew gave another good example from chess theory. Even in brain science, for example, where x is the number of neurons, the relevant number is how many different ways those neurons can be connected, which (given the number of simultaneous connections possible) is greater than not only x to the power of x, but greater than x! (i.e. 1 times 2 times 3 times 4 ... times x). In statistical thermodynamics, entropy involves a count of the number of possible ways a system can be arranged, which is also a much larger number than x for any system of x particles. Etc.
And also by recounting. You can get x = infinity simply by counting the same particle infinite times. You thus don’t need infinite particles for infinity to have a physical referent. That’s in fact how the axiom of infinity is derived in set theory--indeed, they do it by counting nothing (i.e. the empty set) infinite times. Nothing is a physical fact. Counting is a physical fact. Counting indefinitely is a potential physical fact. QED.
N.B. Teapot Atheist replied to this blog. I responded in kind in the following blog: Defining Naturalism II.
Some quick assertions for Andrew & Richard's consideration (keep in mind I'm not extremely well versed in maths or logic):
1. 'Negative' and 'positive' in the physical world are labels. Electrons are 'negatively charged' but it's just a label. I don't mean to say that it's useless, but it's still a label.
2. Magnets have two poles, referred to as 'north' and 'south'. There are good reasons for that, sure. But they are labels.
3. Debt (and liability) does exist, and is properly represented with negative numbers. But there is no such thing as negative money.
4. There is such a thing as negative temperature - but only in relative terms, though (the Celcius scale marks the freezing point of water at zero, and temperatures less than this are 'negative'). Fundamentally there is no such thing as negative heat.
5. My understanding of infinity is that it's a process, a thing you go towards, and therefore not a thing.
LOL... look what I wrote in point 5. I wasn't really thinking was I... ::facepalm::
then there will eventually in actual fact be, at t = omega
Even in a universe of infinite duration, there is no time t=omega, since every value of t (assuming t is representing an ordering of discrete physical events, as otherwise it is a category error to talk about it taking ordinals as values) must be zero or a successor ordinal (since any time after the beginning of the universe must have a predecessor), and omega is a limit ordinal (and therefore has no predecessor).
(It's very important not to confuse concepts like transfinite cardinals (like aleph-null), transfinite ordinals (like omega), the "point(s) at infinity" of the extended real line, the concept of "limit at infinity" (which is just a notational convenience and doesn't involve any actual concept of infinity), and so on.)
As for the long line, you didn't really address that so I'm guessing you didn't look up what it was first. The long line cannot be continuously transformed onto any possible physical line, because it has different topological invariants (which by definition would be preserved by any continuous transform). In particular it is not metrizable, which means that there is no possible real metric that corresponds to a "position" on the line. Any transformation which mapped the long line onto a physically meaningful possible concept will change its properties and therefore invalidate the meaning.
1. 'Negative' and 'positive' in the physical world are labels. Electrons are 'negatively charged' but it's just a label. I don't mean to say that it's useless, but it's still a label.
It's not just a label, it's also a set of properties. For example, electric charge is conserved: if you produce charged particles from uncharged space (as happens in pair production), you always end up with a set of particles whose charges sum to zero when considered as positive and negative integers.
Likewise, you can combine a particle with a charge of +1 (e.g. a proton) with a particle of charge -1 (e.g. an electron) and produce a particle of charge 0 (e.g. a neutron). (This happens in some radioactive elements which decay by electron capture, to take an example which can be observed in the lab.)
So the arithmetic of (static) electric charges follows the same patterns as the arithmetic of positive and negative integers, and thus we can say that integer arithmetic is an abstraction of the behaviour of electric charge, or that the behaviour of electric charge is a physical model of integer arithmetic.
(The assignment of which direction to call "positive" is indeed arbitrary, but that doesn't change the results.)
Pikemann Urge said... I don't mean to say that it's useless, but it's still a label.
...for something that actually exists and obeys the mathematics of negative numbers, i.e. negative numbers are words that refer to those things and the mathematical rules for negative numbers describe how such things behave. Nothing more. Nothing less. That's my point.
(see Andrew's answer above, which reiterates what I just said, with an example)
My understanding of infinity is that it's a process, a thing you go towards, and therefore not a thing.
Infinity is not a process, it's a label for a particular kind of set as defined within set theory (there are in fact several different kinds of infinity, ranked according to "cardinality," some infinities actually being larger than others). Semantically, this describes (at the very least) a potential physical fact (the potential existence of the proposed set) or (at the very most) an actual physical fact (if, e.g., it turns out the universe contains an infinite number of electrons, then an infinite set actually exists).
There is at least one actual physical fact that is infinite, and it's a fact from which infinity is axiomatically constructed in set theory: the number of empty sets there are is always infinite in any definable region (e.g. the number of empty spaces in the solar system we can potentially put a border around or the number of dimensionless points we can potentially count along the edge of a common ruler). Though this derives from the potential fact (of our counting, etc.), the physical fact (that which we would be counting) is actually physically there (not potentially there, but in fact actually there). Hence though counting it up would be a process, it's not as if our counting creates the things counted. The things to be counted are already there whether we count them or not. And those things are infinite in quantity. Hence actually infinite sets exist.
Andrew G. said...
Even in a universe of infinite duration, there is no time t=omega, since every value of t (assuming t is representing an ordering of discrete physical events, as otherwise it is a category error to talk about it taking ordinals as values) must be zero or a successor ordinal (since any time after the beginning of the universe must have a predecessor), and omega is a limit ordinal (and therefore has no predecessor).
You're right, as a position in time-space there would be no omega as thus defined, since that is logically incoherent (and logically impossible things don't exist).
Rather, I should say, to an observer outside the reference frame, all times there will be have occurred (and photons are already in this state: see my blog on the Ontology of Time), hence there will exist (and thus, from the perspective of a photon, there does exist) every t that ever would exist in that universe (which quantity of t's will be infinite, if the time dimension of the universe does not terminate).
The long line cannot be continuously transformed onto any possible physical line, because it has different topological invariants (which by definition would be preserved by any continuous transform). In particular it is not metrizable, which means that there is no possible real metric that corresponds to a "position" on the line.
If it's not possible, it doesn't exist. I don't have to account for the existence of things that can't possibly exist.
Hence I assumed you weren't asking me to do that.
If [the long line is] not possible, it doesn't exist. I don't have to account for the existence of things that can't possibly exist.
Hence I assumed you weren't asking me to do that.
My point is this:
1). The long line does not appear to be something that can possibly "exist" in any concrete sense (it's not an abstraction of anything in the physical universe, even conceptually).
2) The long line certainly is a valid abstract concept, because it can be expressed in terms of well-defined operations starting from the same basic concepts as the rest of mathematics.
So we're back at the point I made in the first comment: it's possible to construct logically consistent, indeed logically necessary, abstract concepts which do not (as far as we know) correspond to anything physical or even potentially physical.
I don't regard this as implying anything "supernatural" or otherwise contrary to naturalism. What I do regard it as implying is that the realm of abstractions is in an important sense not limited by the physical, and we have to bear this in mind when talking about abstracts.
Andrew G. said... The long line does not appear to be something that can possibly "exist" in any concrete sense (it's not an abstraction of anything in the physical universe, even conceptually).
It is potentially. If it can be coherently defined, it can be potentially realized. For example, if the line has length, then that length can be divided into cats (one cat being the length of an average cat, but for which actual cats can be used to compose the line rather than merely measuring it). If it is logically impossible for there to be enough cats to compose the long line, then the long line is logically impossible (and I don't have to account for the existence of logically impossible things). But if it is not logically impossible, necessarily it is logically possible for there to be enough cats to compose the long line, ergo the long line refers to a potential physical fact (that quantity of cats, thus arranged).
Take your pick.
Mathematics is stranger than you imagine :-)
Here's an example. Imagine the real half-line, i.e. starting from some point 0 and continuing infinitely (assuming an infinite universe, or if you prefer we can treat it as the time axis and assume it to be open and unbounded in the future direction).
We can traverse this line as follows: take one step (of one unit length) in the first second, then take one step in the next half-second, then one in the next quarter second, etc. By this method, two seconds suffices to traverse an infinite distance; you can pick any point whatever on the line, and guarantee that you've passed it within some time less than two seconds. (Obviously this isn't physically possible, since your exponentially-increasing speed will eventually run into physical limits, but it's at least conceptually possible.)
On the long line this doesn't work. All increasing sequences on the long line converge; any increasing function, no matter how fast it increases, will at some point reach a limit point with part of the line still not traversed. In fact, we run out of points that we can even logically describe before we exhaust the long line; this in spite of the fact that the long line itself is very simple to define and show to be logically consistent.
It's not really possible to imagine this, because we're not equipped to imagine infinite things other than in the relatively simple senses of unbounded extension and continuity.
I obviously agree (as I've said all along) that the long line can't "exist" in any physical sense. But it is certainly logically valid (requiring nothing beyond standard set theory and topology). You could construct it from cats placed end-to-end only if you had an uncountably large number of cats (not merely an infinite number); this is in fact how it's defined, in terms of the unit half-open interval [0,1) placed end-to-end in a sequence ordered by the ordinal number omega-one (the smallest uncountable ordinal).
It's certainly reasonable to argue that since "an uncountably infinite number of cats" is in some sense an impossibility (since cats are discrete and discrete things are countable) that the long line is equally impossible. But that doesn't make it any less logically valid (and arguably necessary, since it appears in exhaustion proofs - any one-dimensional connected topological manifold is homeomorphic to either the circle, a closed, half-open or open real interval, the open or closed long ray, or the long line).
I think what we're getting to here is that the word "exists" is a bit too slippery when we get too far into abstracts. I suspect most people would say that the natural numbers "exist"; non-mathematicians would probably balk at saying that any transfinite numbers exist; most mathematicians would probably balk at saying that large cardinals (which are independent of standard set theory) exist. Somewhere in between, I would say, there is a grey area where the idea of whether an abstraction "exists" (rather than merely being logically consistent) stops being useful.
Andrew G. said... Mathematics is stranger than you imagine.
The issue is not how strange it is, but simply this: either you can logically construct a given line, or you can't. If you can't, there is no sense in which it exists, even potentially. Period. But if you can, it's logically necessary that that construction can be realized physically (just substitute one cat for every point in that line). QED.
We can traverse this line as follows...
That assumes the passage of time. Don't think in terms of traversing. Think teleporting. Instead of traversing the line, we instantly teleport one cat onto every point on that line. Either we can do that or we can't. If we can't, the line is logically impossible. As can be proved by substitution (any line that can't consist of cats, can't consist of points, either; and by definition a line that can't consist of points, doesn't logically exist; conversely, by the transitive law, any line that can consist of points, can consist of cats).
It's not really possible to imagine this, because we're not equipped to imagine infinite things other than in the relatively simple senses of unbounded extension and continuity.
No arguments from limited imagination, please. That's a fallacy.
I obviously agree (as I've said all along) that the long line can't "exist" in any physical sense. But it is certainly logically valid (requiring nothing beyond standard set theory and topology).
If so, then using set theory we can swap all the points on that line with cats.
If we can't do that, then it can't really have been constructed from set theory in any logically valid way.
But if we can do that, then the line can "exist" in a physical sense.
QED.
You could construct it from cats placed end-to-end only if you had an uncountably large number of cats (not merely an infinite number)
Which is logically possible, i.e. we can potentially have that many cats (they could exist even right now--if the universe is an infinite multiverse of infinite universes, and all are uniformly defined by quantum mechanics, it is necessarily the case that as a result of random quantum fluctuations, there are uncountably many cats--infinitely many in this universe, and as many again in infinitely many other universes).
It's certainly reasonable to argue that since "an uncountably infinite number of cats" is in some sense an impossibility (since cats are discrete and discrete things are countable) that the long line is equally impossible.
Except you can have an infinite amount of discrete things (that's how infinity as a set is constructed: by accumulating discrete instances of the empty set). So there is nothing illogical about having infinite cats.
You explain the line is defined by constructing it out of imaginary things, whether points or segments—I agree; hence they can be logically substituted for cats. What you said is thus a plain statement of the fact I am reporting back to you: when we imagine having all those things to construct the line with, we are picturing a physical state of affairs (the existence of all those things--whether spatial points or cats, it doesn't matter).
In other words, stating what's necessary to construct the line precisely is what it means to say something potentially exists physically. They are synonymous. And thus inseparable. And thus the one reduces to the other. And therefore you don't need anything else to explain what concepts like this refer to. They refer to potential things--and not potential things that are irreducibly mental, but potential things that are reducibly nonmental (like a collection of spatial points, or a really weird collection of cats).
I'm getting the impression that you're out of your depth when talking about transfinites. Do you not understand the important distinction between a countably infinite number of cats and an uncountably infinite number?
If you put a countably infinite number of line segments together end-to-end you just have an infinitely long line like the ordinary real line. The idea of having a line longer than this is one that you can't really expect to handle other than as pure mathematics; and even then, it gets pretty strange.
Andrew G. said... I'm getting the impression that you're out of your depth when talking about transfinites. Do you not understand the important distinction between a countably infinite number of cats and an uncountably infinite number?
Yes. Hence my example directly paralleled Cantor's set theoretic argument for higher cardinality infinities.
If you put a countably infinite number of line segments together end-to-end you just have an infinitely long line like the ordinary real line. The idea of having a line longer than this is one that you can't really expect to handle other than as pure mathematics.
I think you're the one who doesn't understand the logical basis for your own argument here.
Either the long line can be constructed from sets, or it can't.
If it can't, it doesn't exist. Full stop.
If it can, then it is constructed from elements that obey the axioms of set theory.
The axioms of set theory logically entail that any element in a set is exchangeable for any element in any other set, unless that exchange creates a logical inconsistency.
Therefore, if there is any arrangement of sets that constitutes (i.e. defines in set theory) the long line, it is logically necessarily the case that all the elements in those sets can be swapped out for cats.
It doesn't matter what you can imagine or what makes sense to you or what seems logical or impossible. This is a logically necessary fact, all our intuitions be damned.
The only way the conclusion does not follow is if a logical inconsistency is created by swapping any one element, in the construction that is the long line, with a cat.
Can you demonstrate (i.e. formally prove) such an inconsistency?
Holy crap, I really did learn some maths today.
Andrew: Likewise, you can combine a particle with a charge of +1 (e.g. a proton) with a particle of charge -1 (e.g. an electron) and produce a particle of charge 0 (e.g. a neutron).
Okay, I'm fine with that, but electrons don't actually have 'negative' value in the same way that negative numbers do. However, I think you would agree with...
Richard: negative numbers are words that refer to those things and the mathematical rules for negative numbers describe how such things behave. Nothing more. Nothing less. That's my point.
... which sounds pretty fair to me and answers my original objection to negatives and positives.
Swapping elements is only part of the story; a topological space consists of two things: a set, and a collection of subsets (satisfying certain rules) of that set.
The set captures the abstraction "point". The collected subsets are called the "open sets", and in a certain way they capture the abstraction of "close to" or "connected to". Concepts of limits and continuity can be defined in topology without reference to numbers.
For a topological space to be an abstraction of something physical, then it's not, I argue, sufficient merely for the "points" to be equated with physical things; in addition, the topological properties of "close to", "limit", etc., need to behave the same way in the abstraction as they do in the physical world.
The long line has a number of properties that appear to be logically incompatible with a physical world in which everything is described with real quantities. Specifically, all strictly increasing sequences on the long line converge to a limit which is itself on the line (and which is smaller than some other point on the line); even if the sequence is infinitely long and its values increase infinitely quickly. This is simply not possible when dealing with unbounded real-valued quantities; there exist real-valued sequences that increase without limit. (And with bounded quantities, either the limit is not within the bounds, or the limit is a boundary - either way the topology is different from that of the long line.)
In other words, the long line is "too long" to fit inside a universe which is characterized by the real numbers, even though it locally looks like an ordinary line (and in fact you can map its points one-to-one with the real line, but not in a way that preserves the topology).
There are other conflicting properties too. Metric spaces with real-valued coordinates (of any finite number of dimensions) are second-countable, therefore all subspaces of such spaces are second-countable; the long line is not, therefore there does not exist an embedding of the long line into any such metric space.
Andrew G. said... Swapping elements is only part of the story; a topological space consists of two things: a set, and a collection of subsets (satisfying certain rules) of that set.
Which is a set.
That's not two things. It's one thing, in different iterations.
Hence unless a logical inconsistency is created when swapping elements of these sets and subsets with cats, it can logically be done.
So is a logical inconsistency produced?
If yes, prove it.
If not, concede my point.
The set captures the abstraction "point". The collected subsets are called the "open sets", and in a certain way they capture the abstraction of "close to" or "connected to". Concepts of limits and continuity can be defined in topology without reference to numbers.
All irrelevant to my point.
For a topological space to be an abstraction of something physical, then it's not, I argue, sufficient merely for the "points" to be equated with physical things; in addition, the topological properties of "close to", "limit", etc., need to behave the same way in the abstraction as they do in the physical world.
Also irrelevant. All that matters is if the swap is logically possible. If it is, then the long line refers to a potential physical thing. Full stop.
If you wish to further argue that it "also" refers to another thing, some thing that isn't even potentially physical, we can start discussing that. But only after you acknowledge the point that it can refer to a potential physical thing--otherwise, it can't be constructed in Set Theory, and therefore doesn't exist in any sense at all.
The long line has a number of properties that appear to be logically incompatible with a physical world in which everything is described with real quantities. Specifically, all strictly increasing sequences on the long line converge to a limit which is itself on the line (and which is smaller than some other point on the line); even if the sequence is infinitely long and its values increase infinitely quickly. This is simply not possible when dealing with unbounded real-valued quantities; there exist real-valued sequences that increase without limit.
You are confusing actually existing things with potentially existing things.
There is nothing impossible in what you describe, provided we set aside actual limits set by physics, and discuss instead what can potentially be done with a region of space-time if we allow the laws of physics to vary in any logically possible way.
In other words, the long line is "too long" to fit inside a universe which is characterized by the real numbers, even though it locally looks like an ordinary line (and in fact you can map its points one-to-one with the real line, but not in a way that preserves the topology).
Why do you assume a universe has to be characterized by the real numbers?
You seem not to be getting the point. Either no other universe is logically possible (in which case the long line is logically impossible, by reductio, i.e. anything that entails a logically impossible thing is itself logically impossible thereby), or a universe not characterized by the real numbers is logically possible. There is no other option. And if the latter, a logically possible universe is by definition a potential physical thing (i.e. if any universe can exist of such a description, by definition it can exist physically; because conversely, if it can't ever exist physically under any conditions, then by definition it is not a logically possible universe).
OK, I see where we are disagreeing - your definition is wider than I had understood it, so I concede to your argument.
(I'll have to reserve judgement on how useful I think those definitions are; that'll require a lot more thought.)
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