CONTENTS.§1.0 Introduction.
§2.0 Dialogue.
§3.0 Q&A letter (which probably will not get picked).
§1.0 INTRODUCTIONI wish to draw some engagement on the topic of arguments revolving around the applicability of mathematics. There are a number of tasks it seems to me. The first thing is to first of all try to frame or define the issue --- this seems to be really vaguely done at least before a public audience. The second thing is to try to nut out how various world-views understand or make sense of the issue.
There are a number of aspects, which I wish to concentrate on in engaging this matter. Mathematics and physical laws. Mathematics and the mind.
Mathematics and physics is the immediate starting point. There seem to be these laws of nature and they are understandable via the language of mathematics. Mathematics requires a mind to understand it, and minds were not around in a naturalistic universe before biological life spawned, which occurs well beyond its birth. There seem to thus be good reasons to conclude that the mathematically described physical laws, which are (acceptably assumed to be) binding, do not require a mind to simply be in power in governing the universe. On the other hand, it requires and explanation as to why such laws can be in power without any underlying mental structuring, when it requires a mind to even grasp these laws.
Whatever the case might be, it seems to be worthwhile investigating the aspect, how the mind works. Throughout I shall often use the words
Verstand and
Vernunft. I do not know exactly how to coin one of these in English, thus I try to uniformly refer to them in German. Regardless of the language, they refer to successive layers of the mind’s interaction with the world. The most immediate interaction between a person and the world occurs through the senses. The next interface is the Verstand: it takes the material acquired by the senses and structures this with terms and properties. The next deeper interface is the Vernunft (=
reason in engl.), which takes the material of the Verstand tries to assign structure to them --- at this level not with terms but with rules and regularities (loosely speaking).
(For more on this, see my posts in this thread.)Working with this layout, it seems like the mind is only able to recognise patterns and nothing else. It will not recognise a phenomenon, which it is not able to describe in some language or other (be it mathematics, art, some science, or what have you). There is an immediate filtration there taking away the immediate “sting” of specialness (I am sure, that is not a word!). The theologist may nevertheless insist on specialness, as he may only need to point out the existence of any regularity at all.
This roughly describes my starting point. Before proceeding, a dialogue (trialogue?) might provide a more natural transition into the topic (see §2.0).
§2.0 DIALOGUE.Consider A = an agnostic, N = a natural and T = a theist and the following dialogue:
T: the (significantly high) applicability of otherwise mental mathematics to the non-mental physical world, indicates an intelligent structuring the world, and thus an intelligence, which, appealing to a dualistic perspective points to a mind behind the natural world, and thus which must exist beyond it.
N: this is not a valid conclusion. The ‘Verstand’ is only able to grasp those things in nature in accordance to its categories of thought, it is thus blind to those things which it cannot describe via language (in particular, the language of mathematics), and may only perceive order — even randomness does not truly become perceived as orderless, but itself is structured (with terms such as coincidence, risk, chance, etc. as well as with empirical measures, and mathematical structures such as probabilities). Thus you see, the human mind either fails to categorise that which the senses perceive, and thus does not “recognise” it, or else successfully categorises it unsurprisingly with mathematics, scientific terms, etc. The success of mathematics says more the human mind than it does about the universe.
A: Granted. I take your point, N, that there may in fact be some or infact an overwhelmingly dominant amount of the physical realm, which is inaccessible to our understanding with the apparatus of human languages including mathematics. This however does not necessarily negate, that one has to explain why there are phenomenon (and at that, not at all insignificant ones) which lend themselves to descriptions by mathematical structures. I might not be ready to go so far as to conclude, that which T concludes, but I certainly see some worth in the raw material he is starting with: the intelligibility of (at least parts of) the physical universe.
T: A quick overview would suffice here. (1) On the one hand there is the physical realm, with physical objects, and phaenomena. On the other hand the virtual world of linguistic terms (viz. mathematical ones). These linguistic structures are not thought to exist in the world, but provide a thinking mind with material to be able to grasp the world around it: a mind assigns terms, attributes, properties and descriptions to the world and the things it isolates in it. (2) The assigned attributes of these observed objects in the course of interactions and time develop. And tracking this development, it becomes clear to reason (Vernunft) that there are patterns and rules, which seem to happily keep perfect track of the development, regardless of how often and how refined one makes observations. (3) Even though this rule is not at all something, which the mind (Verstand) assigns to the the universe, reason (Vernunft) is able to structure the material of the mind. This second step is something which transcends nature. (4) There is thus a transcendent dimension to the physical world, which is of rational nature.
N: I would agree with the mechanisms you have laid out, but not with the conclusion. You have forgotten, that in the process, all those things which are non-intelligible become filtered out. This is a very relativised experience: the rationality does not in itself have to exist, but is the only thing the mind is able to grasp.
A: Overwhelming one thing with its opposite or trying to drown it thusly, does not warrant one with outright denying its substance. The point remains, there might be an explanation which goes beyond the subject-object level. Just as seeing is not just subject dependent, but there exists explanatory grounds for this which is detached from the subject (namely, the physics of light), the fact that the universe be rationally intelligible at least justifies the inquiry as to an independent explanation.
N: Okay, but some mathematical truths are simply necessary.
T: I agree, things such as arithmetic and combinatorial facts must hold in every world, as soon as one is able to isolate objects, the applicability of the rules of arithmetic is straightforward, an the grounds of explanation, that a human mind can isolated objects (and then count them, pair them off, etc.) is sufficient. But notice these are very static attributes: there is no development. I am talking about phenomenon where one tracks the development of attributes, and to which, broadly logically speaking, one may a priori apply a vast range of mathematical structures and where a posteori it becomes discovered, that a specific structure applies, which is able to predict results, etc.
N: A fair point. Nevertheless, the laws of nature could be anything. If there be any regularity at all to be observed in the universe, there must be some mathematical description.
T: That is not the point. Here I am not talking about that how special these particular laws are, that we have, or that these be special at all. I am talking about the simple issue, that there are laws at all!
N: But that would be my point. I would argue, there is bound to be some degree of order to the universe — one should not be surprise to not observe any order, or we would not be here to observe anything at all. If we permit a vast contingency of possible ways the world could be, there is bound to be some of these possible worlds which admit structure. In this case the sheer range of possibilities provides an explanation for the intelligibility of natural phaenomena.
T: Your approach is to appeal to a multiverse?
N: Not necessarily. I am simply appealing to a broad range of possible worlds.
A: So N's approach is perhaps a combination of multiplying the range of possibilities on the one hand, to allow for an increase in likelihood of structure, whilst appealing to the brain’s behaviour filtering out nonsense, to explain why it only sees structure. Thus one may in this way explain the mathematical nature of nature as well as be able to assert its non transcendence.
N: Yes, in fact, I do not see how appealing to a transcendent mind, would even help as an alternative explanation.
T: Here one is not trying to argue from an intelligent mind to (mathematically) intelligible structures in the universe, but rather the other way around. I am trying to argue, that (A) there is evidence that there is a rational dimension to the natural world and that (B) since taking a purely naturalistic point of view, there can be no intrinsic rationality in nature, this serves as an indicator that there must be some source of this rationality, which goes beyond nature, and which is then echoed in nature. The existence of such a transcendent mind is of course not enough to provide a causal explanation. This provides only one argument in a cumulative case for a theological stance. It would be a further step to then say, on a (mono)theistic viewpoint, it is expected, in creating the universe that God would not simply have brought it into existence, but have structured it and given it some (when not complete) order; what one then observes in nature is in harmony with this view. Again this is the point of the argument we started with. The argument from mathematics would simply warrant this further step.
N: Fine. I would dispute your point (B), by saying that the naturalistic view does not force one into this position, for instance by either denying the intrinsic worth — the intelligibility is always with respect to this subject-object relation between a mind, hell bent on categorising, and the natural world —
T: Right, but one must admit that these structures exist independent of the subject-object interaction. Humans have only been around for ca. 10^7 years, whereas the universe has existed for for ca. 10^10 years. In physics it is a fundamental assumption, that the laws we observe to-day are value spatiotemporally without exception, except perhaps in the singularity. The point still holds, these laws are at least held in the scientific community to existing independently of human existence, even whilst restrained to human language.
N: — or by reducing its speciality, by saying some order must exist when there are a broad range of possible ways the world could have been.
[End of dialogue.]
§3.0 Q&A LETTER.Dear Professor Craig,I listened to the interview on Unbelievable (premier.org.uk) between yourself and Dr Daniel Came about arguments pertaining to the applicability of mathematics to physics. I am personally really passionate about this issue and have to some extent had a look into your views. (I am not sure, but believe, you either deny or do not allow yourself to accept a platonic view, and would say for instance that attributes do not exist, but at most describe objects.)
I realise that Q&A #406 already brushed on this topic, but would still like to pose my question here, as my request is a bit different.
QUESTION 1. I was wondering, if you would be so kind as to provide a layout of strongest positions. As far as I can see, there are at least the following:
- An explanation of the applicability by reducing to the psychological level: the mind filters out everything, which it does not grasp/categorise with language (incl. mathematics), and thus is only capable of seeing those parts of nature which happened to exhibit order.
- An explanation through multiplication of contingent descriptions of the world: the existence of order is contingent (there are many worlds with varying degrees of regularity and chaos) and the form of this order (the specific physical laws) is contingent. (Finally, one may couple this with a Copernicus-type explanation, why we happily happen to enjoy living in such an ordered universe.)
- On the non-naturalistic side (not necessarily theological): the existence of any order at all cries out for an explanation. Denial of this being an happenstance. Denial of this being a purely psychological phaenomenon, since the physical laws are thought to hold for the entire duration of the universe (time and space) before and beyond the existence of human minds.
- The natural laws are accessible to reason (Vernunft) and seem to be strongly undisproved, evidencing their existence; moreover they are not directly physically existent. This indicates (1) a rational structuring of the natural world, which (2) is itself transcendent to nature.
- Theological perspective: this evidences a mind beyond nature and is very compatible with (though does not lead to) the view of a transcendent God creating the universe with regularity and order built in, which has left its trace as accessible to other with-reason-equipped entities.
- Theological perspective: going the other way around (not trying to provide evidence for God), starting with the assumption, that God exists has been helpful in the progress of science, since one then expects order and then looks for it.
- (Naturalist objection: the human mind only ever looks for order regardless.)
- (Theological objection to the objection: this provides a necessary direction, but not the impetus or expectation.)
In particular, is the applicability of mathematics argument one of evidencing an intelligent creator… or is it the other way around, that pragmatically, starting with the assumption, one has the impetus for executing scientific investigation (as in the last three points)?
QUESTION 2. It seemed at some points like the radio-conversation did not quite achieve that level of development, where both sides necessarily built on the strongest claims and replies of the opposing views. Dr Came, for instance, seemed to converge on saying that the laws could be anything given some regularity and seemed to miss the point that the issue is in fact about explaining any regularity or any laws at all. You also seemed to not quite have the chance to really engage Dr Came’s very good response of the psychological aspect of this issue (that namely the mind hones in on and selects out that which is to it aesthetically pleasing/graspable, which thus calls into question the “specialness”). Whilst I have my reactions to this above and below, I am no philosopher and would really love to hear what you would hold would be a good reply to Came's points.
Thanks again for your time and work!
If you have read this far, thanks for the effort. I would appreciate substantive interaction.
