Scientia Salon recently published my article advocating that mathematics is best regarded as a part of science. In reply to “scientism week”, Massimo Pigliucci wrote an article criticising “the return of radical empiricism”. The collision of “scientism week” with “anti-scientism week” generated a lot of energy and comments!
All the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of fact and real existence.
The “relations of ideas” category is taken to include mathematics and logic, where knowledge is “discoverable by the mere operation of thought”, while the “matters of fact” category contains science, where knowledge derives from empirical data.
Kant rejected Hume’s empiricism and sought to establish the primacy of reason. He adopted the term “a priori” for knowledge that does not derive from experience, in contrast to “a posteriori” knowledge which does. A related concept is that of “analytic” statements, which follow from the definitions of the terms, contrasting with “synthetic” statements that describe how the world is.
This notion of a fundamental epistemological divide holds today, and is at the heart of resistance to the idea that mathematics, logic and science are a unified whole.
In reading Pigliucci’s article I agree with much of what he says, but, to me, he seems to miss the main arguments for the essential unity of the different domains of knowledge. I will thus outline how I see the roots of empiricism, and then consider the supposed divide between knowledge “from reasoning” versus knowledge “from observation”.
Let’s start with a primitive animal evolving by natural selection. Such an animal might develop senses to capture information about the world around it. That could help it in many ways, by learning about food or predators or potential mates. The sensory information enters the animal, but what to do with it? Simply feeding the signal to a motor organ might suffice for the simplest of behaviours (moving towards light, say), but as the animal gets more complex it is advantageous to develop more complicated responses to sensory input.
Thus, the animal evolves a neural network between the sensory inputs and the motor outputs. This is a complex wiring pattern that allows multiple input signals to be processed in complex ways, enabling a complex set of responses. The neural network is said to make “a decision” about “what to do” given any set of inputs. Indeed, the whole point of brains is to make such decisions.
In order for highly complex animals to make good decisions, that neural-network decision-making brain must contain knowledge about how the world works. It needs to have internal information that, for example, amounts to knowing that the prey animal bounding away is good to eat, that the lion with sharp teeth would be best avoided, and that the big man with a spear might harm you if he gets angry.
That information is encoded in our neural network in the pattern of synaptic connections between neurons, all hundred thousand billion of them. It gets there by a process with two entwined aspects: a genetic recipe, and a development program. During childhood and throughout our life the brain is bombarded with sensory information from the surrounding world, information which trains the neural network about the world.
But that alone would get you nowhere (as is obvious if you consider streaming the same sensory information at an inert rock). You also need a capability and recipe for what to do with the information, and that is provided by the genes. The genes get that capability from natural selection. Consider random mutations that affect the wiring of the brain: whenever a mutation comes along that causes the network to make better decisions (“better” in terms of survival and reproduction) it will tend to become fixed in the population.
It is reasonably obvious that in order to make good decisions the “world model” contained in the neural network must be a good match to the real world. It needn’t be a perfect match, and it could be a poor match about aspects of the world that had no relevance to survival and reproduction over our evolutionary history (which is likely why we find quantum mechanics very counter-intuitive), but it would have to be a good-enough match to the real world as we experience it in everyday life.
Thus we have a “world model” encoded in a complex neural network, with that network taking in sensory information and making decisions that are then passed to our muscles to act upon.
That world model is derived entirely from contact with the real world, from information attained by sensory organs, and by the natural selection of ideas that best match the real world. All of our intuitive knowledge is encoded in that neural network, and those are the only processes by which it got there.
Two points then follow from how neural networks work. First, in a neural network, knowlege cannot be rigidly compartmentalised. Changes in one connection of a neural network have an influence over a wide area of network. No idea is localised, but rather all ideas are encoded in a distributed web. Thus the ideas all interact and cannot be regarded as independent.
Second, any aspect of the wiring pattern of a neural network can in principle be changed, and thus any of the ideas encoded in it can be changed. It is not the case that knowledge must be hierarchical, starting with a foundation of basic truths, and then with other truths being built on those, in multiple layers where the earlier layers become set and unchangeable. A neural network does not work by layers, it works as a highly interconnected web. Thus there are no “basic” versus “non-basic” truths in the architecture of a neural network.
The ideas of a “world model” thus act as a team, and any part of the team can be changed if the overall result is better. As an analogy, a football coach can swap out any player of his team to produce a better overall performance. “Better performance” here means better modelling of the world.
How is this swapping out achieved? Partially, it is achieved by evolution. If a mutation changes the recipe such that the wiring is changed such that the “world model” performs better, then the mutation will become fixed in the population. But it is also achieved over our childhood and through lifetime experience.
One more concept: The representation of the real world inside the brain will not, of course, be a duplicate of the real world, it will be a model of it. That means it is a set of abstracted ideas that, put together, give a sufficiently good simulation of the world.
What do we mean by an abstracted concept? One can regard the raw sensory input from our visual sensors as a pixel-by-pixel list of photon-arrival events. Obviously this information is unwieldy and useless in its raw form, so it is processed into “concepts”, which are condensed versions of the full information stream. Ideas such as “earth”, “sky”, “tree”, “person”, “distance”, “time”, “weight” are all concepts that compact down external-world completeness into manageable and usable abstractions.
One can only do that if the natural world contains regularities. Examples include the day/night and yearly cycles. Another example is that of species: if we have a concept of “sheep” then we don’t need a representation of each individual sheep. Further examples are what we call “laws of nature” (if you walk off a cliff you’ll fall and go splat) and ideas such as mathematics (enabling you to count sheep) and logic (basic ideas that are necessary to understand the world, even if some of them are so obvious that you don’t consciously think of them).
How do we gain knowledge in our day-to-day life? The method, in its most formal version, is called the scientific method, though this really is just a refinement of the sort of thing we always do to learn things.
If we want to “know” something we simply consult our “world model”, our neural network. That is what it is for, to tell us about how the world works, to enable us to make decisions. So we simply “think about it” and we know the answer. But how do we test whether our “world model” is correct, and that our “thinking about it” is giving a good answer? Easy, we compare the output of “thinking about it” to what does actually happen! If it doesn’t match we update our model, swapping out some aspect and replacing it.
The most rigorous test of the model is by predicting things we don’t already know. So, we predict a property of the world. We obtain whatever information about the world that we need, process that through our “world model”, and then predict, say, that a solar eclipse will happen next week. We can then test whether the prediction works. If it doesn’t there is something wrong with our model (or, possibly with our observed information, so we do it a few times to try to check that it really is the model at fault). So we change the model — swap out a player — and try again.
There is a bit more to it than that, but, in essence, that is the process of science. (One might object that in predicting eclipses we don’t just “think about it”, but also use external aids to thinking such as pen and paper and calculators, and memory stores such as books, but these devices only give us logical consequences of what humans put into them, the external aids are not originating the ideas).
In order to predict that eclipse time we need a good understanding of physics. But to compute anything at all in physics we need maths. And we also need logic, since both maths and physics would fall apart without logic.
So we combine our logic and maths and physics (including all the stuff that we regard as too obvious to need stating explicitly) and combine them all together in our world model and then calculate the prediction.
This process tests the physics, but it also tests the maths and the logic. If the maths and the logic did not hold in the real world then the prediction would turn out false. By that process we arrive at correct maths and logic (where by “correct” I mean best modelling the real world).
That is how we came to the systems of maths and logic that we now have, from the fact that they work in the real world. Of course mathematicians and logicians have since formalised those systems and distilled the mathematical and logical knowledge down into sets of axioms — axioms that capture deep regularities in the way that the world works.
Anyone who doubts that maths and logic are part of the above process, and thus are empirically tested by the above process, is invited to take the negation of axioms of maths and logic (say, the negation of modus ponens, or the negation of Peano’s axioms) and work from there, fully self-consistently, to produce predictions of eclipse timings. It is obvious that that process would not work (you wouldn’t even have counting numbers) and thus that axioms of maths and logic are adopted exactly because they match real-world behaviour.
At this point, I anticipate an objection along the lines that, yes, we swap physical laws in and out, in order to improve the overall model, but we never do that with axioms of maths, do we? We take those as a given.
My reply is that, yes, that is what we do now. But that is only because we’ve already completed the task of arriving at the correct axioms of basic maths (whereas we have not completed the task of arriving at the correct laws of physics). Which just means that the axioms of maths — at least all of those necessary to model day-to-day physical phenomena such as eclipses — were easier to get right! We humans, collectively, have previously been through that process of arriving at correct maths, by some combination of natural selection of genetic recipes for correct maths, experience of the world, and communication of what other humans have worked out.
It follows from the above that axioms of maths and logic and laws of physics all have the same epistemological status. They were all adopted as concepts that model real-world behaviour. There is no other source for any of them (or indeed any other type of knowledge) other than from our contact with the real world, and thus from our experience of how the world behaves. Thus there is no basis for asserting any big epistemological distinction between them.
Going the whole hog
Quine’s famous paper on the synthetic/analytic divide starts by considering the statement that “No unmarried man is married”, which he declares to be “logically true”.
But it this genuinely a priori knowledge, true by logic alone, and entirely distinct from empiricism? In essence it is the basic logical law of non-contradiction, that something cannot be both itself and not-itself. But consider the following:
(1) No unmarried man is married.
(2) No dead cat is alive.
(3) No spin-up electron is spin-down.
At least some interpretations of quantum mechanics hold that (2) can be false, and all interpretations of quantum mechanics hold that (3) can be false. Indeed, the negation of (3) — that an electron can be in a quantum superposition of spin-up and spin-down states — appears to be an empirical fact about how our world actually is.
At the very least, this shows that none of those three statements can simply be assumed as a necessary property of all possible universes, and thus if any are true about our universe then that is an empirical fact about our universe. Indeed, the evidence is that (3) is false and that (1) is true only because unmarried men are large enough (compared to the Planck’s-constant scale of quantum mechanics) that quantum decoherence must always be complete.
This train of thought suggests that radical empiricism does hold and that there are no genuinely a priori truths that are independent of empirical facts. What other justification do we have for declaring the law of non-contradiction, other than that it appears to hold in our world?
The divide between “by reason” and “by observation”
Let’s return to Hume’s fork, and the divide between knowledge arrived at by reasoning, and knowledge arrived at by observation. If we accept what I’ve argued above, does anything remain of this distinction?
Knowledge “by reasoning” would then mean knowledge arrived at by consulting our neural-network model of how things work, complete with its encoded logic and mathematics. That is indeed a very good way of knowing; indeed having the world-model available to consult, in order to use the knowledge as an input to decision making, is exactly why evolution has equipped us with brains.
Thus, in order to know how many sheep we would have if we combine our own flock of 15 sheep with our brother’s flock of 17, we do not have to merge the flocks and then count them, we can simply reason it.
The brain is the product of a long history of contact with the world (over evolutionary time and over our own development from an embryo) and has distilled all of that observational knowledge into a handy ready-reckoner that we can consult to get an answer without actually having to do the observation.
Thus the knowledge that the combined flock will be 32 sheep is then not empirical, since it was attained by reasoning rather than observation. Is that, then, a priori knowledge?
Now let’s consider that we are standing on the edge of a cliff. We know that if we step forward we will fall and go splat. Yet, we have not observed that event. Our knowledge of it again comes from reasoning, from consulting our ready-reckoner brain. The knowledge here is again about how the world works, and is again a product of a long history of experience of the world.
I don’t see any epistemological difference between this knowledge about the effect of gravity, and the above knowledge that 15 + 17 = 32. The first is from laws of physics and the second from axioms of maths. Yet, traditionally, maths would go in the “a priori” bin and our knowledge of gravity in the “a posteriori” bin.
Now let’s suppose that we have encountered the big man with the spear. We know that if we go and thump him on the nose he might be angry and thus that thumping him may well be a bad idea. Yet, we have not observed that man before, and have not observed what happens when he is thumped on the nose. Therefore we have no empirical knowledge of him. What we’ve done is consult our ready-reckoner brain, and the fact that people get angry when thumped on the nose is one of those regularities of nature that has been encoded, by long experience of nature, in the brain.
So, our knowlege that thumping the guy is a bad idea is “by reasoning” and not by observation. It is “by reasoning” in exactly the same way that knowing how to add the numbers 15 and 17 is “by reasoning”, namely it is done by thinking with our world-model brain — though that world-model brain is of course entirely a product of contact with the world.
Now let’s suppose we encounter a new type of animal that we didn’t know about and which behaves weirdly, perhaps it hops whereas we Europeans are used to animals that walk. The first time you see it you learn — empirically — that it hops. From then on, though, the knowledge that this animal hops is incorporated in the world model. So if you later see another animal that looks to be of the same species you know that it hops, even before you have seen that animal hopping. The knowledge of the first animal hopping was observational and your knowlege that other animals of the same species then hop is from reason.
How let’s turn to observational physics, the very epitome of empirical science. And physics has empirically observed the Higgs’ Boson. Err, has it? Have you ever seen a Higgs Boson? Have you ever seen a picture of it?
What you may have seen is a piece of paper with plot on it, a plot that shows a line computed from a model, and compares it with data points with error bars. Those data points are records of detections, not of the Higgs’ Boson directly, but of various decay products that we theorise to occur, if the Higgs does exist. This involves some empiricism, but it also involves a huge amount of theorising and analytic maths-like deduction from our overall world model.
Indeed, the train of deductive reasoning is so long, and involves so much reference to other parts of the world model, that anyone not trained in physics will struggle to assimilate it. Thus the discovery of the Higgs is roughly 5% new observation and 95% deductive reasoning from the world model. Of course all of that world model is fashioned, ultimately, out of the contact with the empirical world — but that is just as much true of our mathematics.
As a last example, take the temperature at the centre of the sun. That is surely an empirical matter, and we can state as a fact that the temperature is 16 million Kelvin (subject to the usual scientific caveats about error bars and provisionality). Yet we don’t know that directly. What we do is measure lots of other aspects of the sun, such as its mass (which is also not known directly, but deduced from other information that ultimately traces back to empirical data) and then deduce the temperature of the sun by feeding everything into a model.
The lesson I take from the above discussion is that all knowledge includes both observation and reason, both a priori and a posteriori elements, and that any distinction between the two is unimportant.
Yes, you can make a distinction between the animal that you have personally observed to hop and its conspecific that you merely presume to hop. But, really, since all of our knowledge is such an entwined mixture of both reason and observation, insisting on such a distinction seems perverse.
One can distinguish between proximate empirical knowlege, where you have recently observed something, and distant empirical knowlege that is now just stored abstractly in your ready-reckoner device, ready to be consulted when needed, but I don’t see why that distinction matters much from the point of view of fundamental epistemology, especially since we know nothing at all except as processed through the world model in our ready reckoner. Without that we would not even “see” a kangaroo, we’d just have a list of photon-arrival event times from each photo-sensitive cell in our eyes.
Everything we know comes ultimately from contact with the empirical world, but is so entwined with our modelling and theorising about the world that the distinction between knowledge “by reasoning” and knowledge “by observation” simply dissolves.
Trying to establish the primacy of either empiricism or rationality as the basis of knowledge will not work. At the basis of knowledge is, instead, Darwin’s dangerous idea of natural selection. Darwinian evolution fashioned our reasoning, and did so by contact with the empirical world. Reasoning always was the product of the world we experience, and the two aspects of knowledge “by reasoning” and “by observation” always were inseparably entwined right from their origins in our evolutionary past.