Is Mathematics Discovered or Invented? A Novel Resolution.
In this essay, I will present an argument that the answer to the age-old debate of whether mathematics is discovered or invented is that it is neither!
Read on to see what I mean.
I have found that whenever there is virtually endless debate between two opposite sides on an issue, it is because each side has both strong and weak points, but neither fully captures the “essence” of the position that allows one to properly understand the issue at hand. Each side is partially right and partially wrong, and that is why neither side can conclusively defeat the other.
I believe this is also the case for the two age-old positions on the question of whether mathematics is discovered or invented.
I will first lay out these positions before proposing a resolution that is meant to integrate the best of both.
The objects and structures of mathematics, such as sets, numbers, various abstract spaces and so on, are not physical, and that would suggest that mathematics is invented.
There are mathematicians who believe this to be the case and have made it an essential aspect of their research methodology. They are called intuitionists, and if they are right, then mathematics is definitely invented.
But if mathematics is invented, how is it that it is so remarkably consistent, that it gives rise to such a remarkably coherent edifice, and that often we seem to “discover” hidden connections not only between different parts of a given branch of mathematics, but also between different branches of mathematics, and even between mathematics and the sciences, in the sense that mathematics is remarkably (and remarkably consistently) useful in modeling aspects of the real world?
How is it that this “invention”, if that is what it is, seems to both predate us and is almost certain to exist long after we are gone? To consider something as an invention of ours presupposes our existence, and yet mathematics does not seem to depend on our existence.
On a more subjective level, why does it so consistently feel like a discovery when we recognize new connections, or a new way of proving something?
All these considerations would suggest that mathematics is discovered. But, as mentioned, the objects and structures of mathematics are not physical, so exactly what kind of things do we discover? Is there some kind of Platonic plane of reality “out there” where these objects exist?
Some people do believe this, they are called Platonists, and if they are right, then mathematics is definitely discovered.
But how can we ever demonstrate that such a plane exists? What does it even mean to claim that such a plane exists? How would the claim of its existence be any different from the claim that Heaven and Hell, or for that matter, Hades, Valhalla, Kur, Trailokya or Mictlán exist?
If there is no difference, then it is purely a matter of faith. But if the existence of a Platonic plane is purely a matter of faith, we haven’t really settled the debate, but only adopted one position based on faith without evidence.
One potential way to overcome this problem is to consider that Nature itself is mathematics! This is called the mathematical universe hypothesis (MUH), and if it is true, then new insights into mathematics should be considered discoveries because they are now literally a kind of scientific discovery.
But the MUH has the same problem as before: how can we demonstrate whether it is true or not?
Proponents of the hypothesis point to the phenomenal success of mathematics in representing virtually all aspects of reality in the sciences. Indeed, that is what inspired the hypothesis in the first place.
But it must be kept in mind, the approach in math and science to get closer to truth, and generally the approach to avoiding confirmation bias, is to focus not so much on supporting evidence, but on evidence that supports the negation of one’s position.
How strong is the evidence that the MUH might not be true?
Well, for one thing, the fraction of mathematical structures and ideas that are used to represent our reality is minuscule compared to mathematics as a whole. Most of mathematics has no known relation to our reality!
For another, the mathematical concepts of mass, charge and related quantities seem to somehow fall short of the physical and metaphysical concepts, in the sense that accounting for all the laws and symmetries they obey at best exhausts the description of what they do, not what they are.
Thirdly, there is a branch of mathematics called inconsistent mathematics, which tolerates certain limited kinds of inconsistencies without breaking down. In the abstract, it is easy to contain or “quarantine” this branch of mathematics from the others. But if nature is mathematics, then inconsistent math, to the extent that it is reality, should infect everything else with its inconsistencies.
Most importantly, in my opinion, there is a certain extremely subtle bias at play in how we conceptualize the role of math in science: we specifically look for mathematical explanations to play the role of “mediators of understanding” of nature.
This is in stark contrast to how things were before Galileo and Newton, before science began to be mathematized. In those earlier times, people followed the physics of Aristotle, and to an Aristotelian, a mathematical explanation would not be any explanation at all if it failed to explain anything about the conceptual foundation of their physics, namely Aristotle's four causes.
So, it could be that mathematics only plays the role of “mediator of understanding” of reality for us because we are blind to a whole host of other and non-mathematical ways of understanding reality. If that is true, it would considerably weaken the motivation for considering nature itself to be mathematics.
The bottom line is, it does not take much thought to realize that the MUH is not as well-supported as its proponents would have you believe. And thus we are back to the question of what it even means for mathematics to be discovered.
I went through all this because it is one thing to just say that each side has strong and weak points, but something else entirely to describe what these points actually are.
To summarize:
- The strength of the “invention” side is that mathematical entities are unphysical (unless the MUH holds). Its weakness is that it makes the remarkable coherence and consistency of mathematics and its seeming independence from our existence, as well as its central role and utility in representing our reality, seem like a miracle.
- The strength of the “discovery” side is that it captures all the ways in which mathematical progress does indeed seem like discovery. Its weakness is that we have nothing backed by evidence on which to ground the idea of discovery for mathematics.
The process that I envision as an alternative to both the “invention” and “discovery” processes can be likened to planting and harvesting a seed plant:
We choose the seeds which are to be planted, and we also choose from various contextual parameter values like the soil, shade, watering schedule and amount, and so on, those which it is going to be subject to.
But once we are done with that, the plant does the rest of the work on its own. Our only role, once the plant has grown, which is to say, once it has developed the kinds of structures that could only develop from that seed and under those contextual parameter values, or simply context, is to harvest it.
Notice that this is one process but two steps (at least in the widest sense).
- The Seeding Step is the analog of the formulation of the axioms. The determination of the contextual parameter values is the analog of the determination of the scope of the axioms and things like the determination of the basic formal language in which the axioms are to be formulated (e.g. first order logic) and similar considerations. Notice that this step comes closest to invention.
- The Harvesting Step is the analog of everything that is related to deriving consequences from the axioms: defining additional structures that build on the axioms, making conjectures, and proving theorems. Notice that this step comes closest to discovery.
So, once we have “seeded” the axioms, which we can determine as we wish, the mathematical structures that “grow” out of them are set. It is just left for us to find out what they are.
Some axiom sets lead to richer mathematical structures than others, just as some seeds grow into more magnificent plants than others. An inconsistent set of axioms would correspond to a seed which fails to produce a plant, a dead seed.
The axiom set in combination with the context, rather than any Platonic plane, serves as the ground for the mathematical structures that grow out of it.
The fact that certain mathematical relations held in our reality long before the arrival of any life on earth or any life at all is to me evidence that the Seeding and Harvesting process has a pluralistic nature of which one is epistemological (pertaining to our knowledge) and another physical.
The idea behind the latter is that Nature itself may plant seeds out of which physical structures grow which behave in a mathematical way (I am NOT supposing any kind of intelligence here, the step reflects simply “nature taking its course” as it does with anything else).
The physical Ur-seed would be the big bang, and a physical seed relevant to our own existence would be the local conditions just before our solar system began forming, in terms of the incidental distribution of matter and energy. The context would be determined by the applicable laws of physics, the particular values of physical constants and the possible influence of fields originating from elsewhere.
Harvesting in this case then corresponds to the conditions of the relevant aspect of nature at any moment after the Seeding was completed.
In contrast, the formulation of axioms (and determination of context) for a given branch of mathematics, as well as the harvesting step, are all epistemological. Even the grounding of the mathematical structures on the seed and context in this case, reflecting a relation between two epistemological objects, would seem to have to be so.
Purely epistemological structures can still be objective, and the dual relationship with reality puts weight behind the epistemological grounding and makes it concrete, insofar that we can find a counterpart in reality itself, of which our epistemological structure would then be a representation. In the epistemological case, harvesting would still be akin to discovery.
But I would avoid the word “discovery”.
First, it comes with all the baggage that stems from discoveries in our real world, which are very unlike the analogous process in mathematics.
Second, I would avoid it because it conveys too much the idea of a complete process, as opposed to a step in a process.
I would avoid the word “invention” for similar reasons.
Finally, I believe that a Seeding and Harvesting process applies not only to mathematics and nature but to every product of our imagination. For example, consider the age-old question of whether art is invented or discovered. Or literature, music, architecture, programming languages and so on.
In the case of art, for instance, Seeding may correspond to the determination of the core artistic ideas to be expressed, together with the determination of which medium, tools and aesthetic or other rules to use, all of which create a context against which the finished work of art grows out of the core artistic ideas.
To conclude, if my proposed conceptualization of mathematical progress is correct, then mathematics was neither invented nor discovered but subject to a two-step process which combines aspects of each, mathematics is what I call seeded and harvested.
