Discovery and Invention Part I: Distinctions and Notations

In this three part (probably) series, I’m going to look at the notions of invention and discovery as they relate to how we think about mathematics and logic. In this first post, I’m going to set up the distinction between discovery and invention as I see it, and then talk about whether systems of notation for logic and math are discovered or invented. The second part will look at how our philosophical views affect whether we think math is invented or discovered, and the third will do something similar for logic. (links will be added as the posts are finished)

Discovery vs. Invention

In science (especially), there are some sorts of things that are clearly discoveries: “We discovered this weird fish at the bottom of the arctic ocean”, “We discovered a molecule that kills cancer cells”, but “Someone invented an artificial heart”; “Someone invented a more efficient fuel injector”.

It seems odd to say you invented a molecule unless you really did construct it for the first time in a lab, and it sounds odd to say someone discovered a mechanical device if they were the first to construct or design that device.

What it comes down to, broadly speaking, is whether something was out there already or whether it’s been created for the first time (or at least independently of similar inventions). What first comes to mind when someone mentions inventions are probably (types of – I’ll suppress this qualifier from now on) physical objects — space shuttles, the wheel — or materials — plastics, alloys. But we also sometimes talk of people inventing characters, musical themes, or abstract systems (government structures for example).

In all of those cases I would class the objects of invention as artifacts. In this context I will take a artifact to be an abstract or physical object (or type of object) that is dependent on minds (human, animal, or alien) for its initial existence. (Depending on exactly how we want to cash out “invention” this definition may turn out to be circular, but it good enough for what I want to say here.)

The objects of discovery, on the other hand, are mind independent —  they would have been there whether or not we were around to find them. Planets, quarks and fish are the sorts of things that can be discovered but not invented.

That said, we also speak of archaeological discoveries and discoveries of maps, or even previous inventions. In all of these cases what’s being discovered is something created by (i.e. dependent on), human minds, albeit minds far removed from our own. This suggests a broader reading of meaning of ‘discover’ along the lines of ‘find something previously unknown (to the discoverer)’. So then  we might say that to discover something is to find something not known for a significant time (if ever). In any case, math and logic are unlikely to fall into the category of things created by temporally and cognitively distant minds and found again in modern times, so I’ll spend no more time on analysis here.

There are two further assumptions I’m going to make about invention and discovery. Either could be denied, but I find them plausible. First, I take invention and discovery to be mutually exclusive in the following sense: one can’t be the discoverer and inventor of the same object. Again, a previous invention can be discovered, but the original object is an invention not a discovery.

Second, I don’t take invention and discovery to be exhaustive. There are things we create that aren’t inventions — new thoughts, new combinations of words or Lego, works of art. . . It would also, I think, be odd to think of people discovering trees or rain — ubiquitous natural occurrences don’t seem like the sort of thing we, as people on Earth, could ‘discover’.

Systems of Notation

Now that we have some sort of handle on the difference between discovery and invention, the first thing I want to get out of the way is the fairly straightforward case of notation systems. It is clear to me that these are invented. In the case of logic, we can (and do) represent the same content in Frege notation, Polish notation and standard Peano-Russell notation. Even within the latter we might represent conjunction as or as without loss or change of meaning.

The case of mathematics is similar. When doing calculus, we’re generally happy to move back and forth between Newton’s notation () and Leibniz’s notation () based on convenience. Likewise, we can express numbers using Roman or western Arabic numerals depending on what century we’re living in.

It is also clear that had Frege never been born, Frege notation (probably) wouldn’t have existed, which is to say that the existence of particular notation systems is dependent on minds (though generally not one particular mind).

One way to think about mind-independence is using couterfactuals: if there hadn’t been human’s, would there be logical notation systems? The answer, surely, is ‘no’.

It should be clear then, that notation systems aren’t discovered. Are they invented? We generally think of inventions as new creations, perhaps requiring some certain level of novelty and/or usefulness. If you’ve ever seen Frege notation, you have to agree that it’s novel, and if you’ve ever used the calculus, you’d probably find that having some system of notation is indispensable.

This isn’t a watertight argument, but it seems pretty clear that notation systems for math and logic are invented. In Part II, I’ll look at what it means to think of math as invented or discovered.