3

I'm not asking whether mathematics is discovered or invented, rather whether being theist implies/strengthens/related to the view that it is discovered, and vice versa.

For example I came across an article by an intelligent design proponent who was defending the view that mathematics is discovered.

So I want to know how are these two issues related: theism and the nature of mathematics.

5
  • An oddball idea in this vein is Cornelius van Til's claim that Trinitarianism is a solution to the one-over-many problem, with whatever ramifications for the ontology of plurality that that would imply. Cantor equated the universe of all numbers not with a static container, but a "living God" (more precisely, an actus purus, a being of pure action), with the quasi-realist consequence that we discover the thoughts of God as we explore mathematics (though why those would be uniquely divine thoughts isn't quite apparent). Commented Aug 16, 2023 at 15:38
  • Dummett used (mathematical) intuitionism as a base to further explore verificationist semantics, and then applied that to theism in his Gifford lectures. fwiw, he didn't actually think too highly of the constructivist project. other than that, your question is a little broad- there are many versions of theism
    – emesupap
    Commented Aug 16, 2023 at 15:51
  • If the theism is the traditional monotheism where God is the author of everything then he is the author of math in particular. This makes it easy to argue that math is discovered, by humans that is. God, of course, is then its inventor. This is not logically ironclad, and one can find creative ways out of it, but that would be much harder. Compare to the Euthyphro dilemma for morality in place of math.
    – Conifold
    Commented Aug 16, 2023 at 16:34
  • 2
    I suppose you're really talking about math theories, not some mundane exercises or practices from established known theories. Anything theory related perhaps is intimately related to theology or theism simply by a straight glimpse of their common linguistic prefix theo/the from etymology... Commented Aug 17, 2023 at 6:21
  • Everything is related to theism.
    – Scott Rowe
    Commented Aug 17, 2023 at 10:32

3 Answers 3

1

The SEP article on Arabic and Islamic philosophy of mathematics discusses Avicenna's response to one version of mathematical realism:

[Mathematical objects] can be defined (or conceived) with no reference to matter or material beings. On the other hand, everything that is separate in definition (or in mind) is separate in existence. Therefore, the argument concludes, mathematical objects are separate in existence. They exist as fully separate beings which have no association with matter or material beings (Avicenna [MPh]: chap. VII.2, sec. 5). However, Avicenna finds this argument wanting. He argues that there is a difference between (a) defining (or conceiving) something without the condition of materiality and (b) defining (or conceiving) something with the condition of immateriality. He says that mathematical objects are separate in definition only in the sense of (a). But the second premise of the argument under discussion is true only if the separation in definition is considered in the sense of (b). The mere fact that something can be defined without the condition of materiality does not entail that that thing can exist in the extramental realm fully separate from matter. But mathematical objects cannot be defined with the condition of immateriality. It is not plausible to suppose immateriality as an essential component of the definitions of mathematical objects, or so Avicenna claims.

He apparently ends up with an empiricism-minded realism, however:

Most Muslim thinkers who have talked about the epistemology of mathematical concepts believe that these concepts are formed through some cognitive mechanisms whose first input is the data we receive through our external senses. The details of such mechanisms are spelled out in different ways by different philosophers, depending on their general picture of human cognitive psychology. For instance, Avicenna puts forward a thought experiment showing that no mathematical concepts can be grasped in the absence of sense perception (Avicenna [MPh], chap. VII.3, sec. 1; Zarepour 2019: sec. 5; 2021, sec. 3). This indicates that Avicenna endorses some sort of concept empiricism about mathematics.

Anderson[72] discusses the "freedom of the mathematician" as a question for Aristotle and Thomas Aquinas:

[Quoting Aquinas] "The mathematician and the natural philosopher treat the same things, i. e., points, and lines, and surfaces, and things of this sort, but not ill the same way. For the mathematician does not treat these things insofar as each of them is a boundary of a natural body, nor does he consider those things which belong to them insofar as they are the boundaries of a natural body. But this is the way in which natural science treats them .... Because the mathematician does not consider lines and points, and surfaces, and things of this sort, quantities and their accidents, insofar as they are the boundaries of a natural body, he is said to abstract from sensible and natural matter.

Anderson partly summarizes his interpretation of Aquinas thus:

... the imagination has an especially important role in mathematics for Aquinas-a role which, as we have said, is never mentioned by the Stagirite. For, in addition to providing a stable image from which the universal can be abstracted (this it does in all abstraction) in mathematics it furnishes to the intellect perfectly appropriate individual mathematicals, which simply cannot be found in nature, individuals from which the mathematical essence can then be abstracted. The direct senses are able to supply an appropriate object for the abstraction of physical essences; for the intellect's abstraction the imagination simply provides a stability in the changing objects grasped by sense. But the direct senses themselves cannot provide a perfectly appropriate object for abstraction of mathematical essences, for mathematical objects as such are not attainable by these senses. Rather the imagination, through its abstraction discussed above, provides the proper object, the suitable individual mathematical quantity, from which the mathematical essence can be abstracted.

Byl[16?] starts out by asking:

What does mathematics have to do with God? Not much, according to most people outside of this association. It is, therefore, quite surprising how often non-theists, when discussing the foundations of mathematics, refer to God. Many contemporary philosophers of science perceive theism to be the basis for classical mathematics and mathematical realism, both of which are therefore found to be objectionable.

I don't know who Byl is referring to, exactly, though Georg Cantor and Kurt Gödel were (more or less) mathematical realists who put God, as an actus purus or ens realissimum, at the apex of their worldview (or their general hopes, anyway). (Mueckenheim[19] includes a quote from Cantor (page 24) in which the phrase "actus purus" is used; c.f. Cantor's thematic debt to Augustine (see Drozdek[95]). For a neo-Anselmian thesis, see Steinhart[09].)

Even L. E. J. Brouwer, who was not an ante rem realist, flirted with a notion that echoes talk of mathematical gods or at least demigods (so to speak), the notion of a "freely creating mathematical subject" whose choice sequences piecemeal determine the constitution of a continuum a priori. A brief synopsis of Brouwer's life in the SEP doesn't emphasize any religious inclinations he might have had (if any), however.

I would tentatively suggest, then, that theism is weighted towards neither the discovery picture nor the invention picture, but might be better styled an intermediary or outlier standpoint with respect to the given question. We human mathematicians might "discover" the thoughts of God, on the above views, but conceptually, God's relationship to Its own thoughts would be less happenstance-based and more proactive, i.e. inventive. For example, if even arithmetic is "synthetic a priori" at least in the way of e.g., "1221 = 46005119909369701466112," being deniable without "contradiction," then one wonders whether a god of numbers would have the power to decide the otherwise contingent laws of arithmetic: to invent arithmetic, in other words.

1
  • 1
    That's the direction I was leaning. Perhaps the reason God comes up for people is sort of like how our eyes have a blind spot: it is just how our eyes happen to be. We can't figure out God as a result of this lacuna of mental functioning. The blind spot in the eye doesn't mean something isn't there to see, or that something is there, it simply means that we can't see in that spot. Mentally, there is something we can't see in to, and that doesn't mean God is there, and doesn't mean God isn't there. It means that's how our mental functioning is.
    – Scott Rowe
    Commented Aug 17, 2023 at 10:27
1

Platonism is 'traditionally' a 'mild' problem for theism but not a 'large' one. (one doesn't see too many atheistic arguments from platonism...)

The crux of the problem is this:

(i) via perfect being theology (anselm), we deduce that a being who grounds mathematical objects is greater than one who doesnt. (see also Why have philosophers historically defined God as omni-benevolent?). So God grounds mathematical objects. Of course, other reasons why God might ground mathematical objects might be given (Feser on Augustine).

(ii) mathematical objects are largely thought to be necessary, hence they must exist violating perhaps God's freedom of will to create.

For more, see perhaps van Inwagen, Craig, Plantinga, etc on the notion of divine aseity. For a neat summary of the modern issues see : https://plato.stanford.edu/entries/god-necessary-being . As for historical precursors, see Descarte on theistic voluntarism, or Augustine/Aquinas on the identity of Gods mental states with mathematical objects. The scholastics in general explored this issue quite thorougly.

1

Georg Cantor's attempts to ground his theory of infinite cardinalities in revealed theology have famously made him the butt of many jokes; see e.g., the contemporary caricature reproduced in Dauben's book. But since a majority of modern mathematicians (like other academics) are secular, and on the other hand tend to hold Platonist views concerning at least the real number line, the relation seems to be the opposite of what your question is trying to suggest.

Meanwhile, if one takes a broader view of theology, one detects some characteristics of revealed theology in modern mathematical Platonism itself. You have the revelation date of around 1870 when Weierstrass taught us the True Rigor. Carl Boyer's description of Cantor, Dedekind, and Weierstrass as the "great triumvirate" has a political rather than a theological ring to it, but the implied attitude of reverent admiration and adoration is similar.

The widespread belief (faith?) that the real number line possesses a "standard model" (a.k.a., "intended interpretation") is so deep -seated as to resemble theological dogma. In my own experience, attempts to suggest, following recent work, that the theory of real numbers can accommodate the presence of infinitesimals are sometimes met with reactions ranging from incomprehension to bewilderment.

5
  • Paras 2, 3 are fine. Para1 not so much. Your secular arises in a context where people wish to separate an overbearing church, usually catholic, from mundane life, in short an entirely expedient political vector. But then you repurpose it to philosophical, ontological and ultimately theological ends. In short it's a very Christian world view See Rabbi Sacks telling atheist Dawkins Richard you are an atheist But a Christian atheist [22 mins]. Ie you're misusing the natural polysemy of 'secular' from non religious through irreligious to anti religious.
    – Rushi
    Commented Aug 17, 2023 at 6:45
  • Should add: The last sentence is intriguing. I would wish an expansion on the details!
    – Rushi
    Commented Aug 17, 2023 at 6:55
  • @Rushi: I added a link for your perusal. Commented Aug 17, 2023 at 6:58
  • Tnx. But I was asking not for the math details but the incomprehension/bewilderment reactions. Eg as Kronecker said God made the natural numbers... Or Gordon who called AC "Not math but theology" Or Wittgenstein who said "If Hilbert can call (Cantor) heaven why can't we call it a joke?" etc etc
    – Rushi
    Commented Aug 17, 2023 at 7:23
  • @Rushi, as you know, mathematicians today are trained in the Weierstrassian paradigm, and are often influenced by Cantor's (and Russell's) virulent anti-infinitesimal attitudes that have percolated through several generations. Mathematicians tend to interpret the adjective "real" in the expression "real number" in its generic meaning of "truly existing" somewhere out there, and similarly for the real line. Cantor and Weierstrass would have balked at the idea that one could find infinitesimals in the real line, and modern mathematicians tend to follow suit, though some are open to such ideas. Commented Aug 17, 2023 at 7:31

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .