Is the Law a Thing?

Material objects are easy to localize, as we may picture them as existing in space and time. This is equivalent to describing one thing (the object) in relation to other things (space and time).

In particular, an activated neural pathway is a thing in space and time. But how about the law associated with this thing? Where does the law reside? Say, I imagine in my mind the color blue. Where does this imagination/law reside? We expect this to be the law interpretation that the watcher associates to the corresponding neural pathway, and we would not be averse to accepting that the law is ‘resident in’ or in some way locally associated with the pathway. We would not want to say that the imagination of blue is somewhere out there in some surreal space, and that the activated neural pathway somehow discovers that imagination. The imagination is created/invented, not discovered. Similarly, a poem is invented, not discovered. Another example; say I recall that it rained yesterday. There is a neural pathway storing the law that ‘it rained yesterday’. We would not want to assert that this memory lives in some outer abstract space, and we discover it; rather, the mind created the memory. Or the neural pathway that discovers the resident law of the concept of ‘mass’. Next example, say the number five. It is an abstraction—a law—associated with the nerve path for the number five, and we are happy to accept that this law ‘lives in’ the path.

Now it gets more interesting. Let us think of relations between numbers. As one out of innumerably many examples of the magic of number theory, consider the infamous 3n+1 problem, which is stated as follows. Start with a natural number N. If it is even, divide it by 2. If it is odd, multiply it by 3 and add 1. Repeat the same algorithm with the resulting number. For every number that has been tested, the process always ends in the cycle 4, 2,1,4.

But till today there is no proof for this for arbitrary N; and it has been labelled as one of the toughest problems in mathematics, for which mathematics is not yet ready [the Collatz conjecture]! [5].

More relevant for us here is the randomness apparent in the 3n+1 sequences of different numbers. Here are a few examples:

5, 16, 8,4, 2, 1

7, 22, 11,34, 17, 52, 26, 13,40, 20, 10, 5,...

25, 76, 38, 19, 58,29, 88,44, 22, 11,...

26, 13,...

28, 14, 7,...

These are all small modestly sized sequences. But now look at the sequence for 27: 27, 82, 41, 124, 62, 31, 94,47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593,1780, 890,445,1336, 668, 334, 167, 502,251, 754, 377,1132, 566, 283, 850,425, 1276, 638, 319, 958,479, 1438,719, 2158,1079, 3238, 1619,4858, 2429, 7288, 3644,1822,911,2734,1367,4102,2051, 6154, 3077, 9232,4616,2308,1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1.

This sequence has 111 steps! What on earth suddenly happened between 26 and 28? We do not understand this, in fact. Now, there of course must have been a neural pathway established in the brain when we worked out the above sequence; and a similar pathway in the computer, if we used a program. But this sequence is so objective and universal in nature, and agreed upon by everyone, that it is impossible to believe that the neural pathway created/invented this sequence, and that what we see above is the subjective law interpretation of the thing. The sequence very much seems to have a life of its own, showing no sign of any human involvement, but rather belonging to the world of numbers, which exists somewhere out there, and the neural pathway only discovers it, and then stores it. This same Platonic feature is of course true of all numbers, and of all mathematics.

Now we are in trouble.

Because there seems to be no evidence from neuroscience to suggest that the connections which represent thought, the color blue, the rain yesterday, or the number five, are of a fundamentally different nature in construction, as compared to the connections which represent a sequence such as that for 27. Yet we very much believe that a thought is a subjective law which is resident in the thing (the neural pathway), but the number sequence is an objective law not resident in the thing, but only represented by the thing, and resident in a Platonic world.

Considering the diversity and subjectivity in the connectomes of different people, how are we to resolve this apparent conflict between the subjectivity of such thinglaws as thoughts and feelings, and the objectivity of thing-laws such as mathematics? Is there a world of mathematics somewhere, which the neural pathways discover, when we think mathematics? No. Because that belief in the Platonic ‘somewhere’ of mathematics is nothing short of supernatural. To believe that mathematics has a world of its own is a bit like believing in ghosts. Nobody has seen ghosts, yet some people are sure they exist. Rather, to resolve the aforementioned conflict, we make the bold proposal that a law is also a thing; it is the same as the thing which it represents. The difference between an abstract law and the thing which codes for it is an illusory difference. This is true as much in the neural pathways in the brain, as in the material world outside. Mathematics resides in the things of the outside world, and the same thing-law association is represented in neural pathways. Thus in the material world, we may view the 3n+1 sequence of 27 as follows: get a huge pile of a very large number of bricks. Pick say 27 of them. Then add 55 more to this lot to make a total of 82. Then halve the lot to 41. Then triple this lot and add another. And so on. And after 111 steps we will be left with just one brick. If mathematicians one day discover the proof of the 3n+1 conjecture, then where is the thing aspect of this proof? In the mind, the thing is the neural pathway that corresponds to this law. How are we to see the proof of the 3n+1 conjecture in the pile of bricks, without having to physically test it again and again with N bricks, for different N? We believe the proof is ‘in the bricks’, but in a complex emergent universe such as ours, this is not apparent. But if we investigate into deeper and deeper reductionist layers of physical reality [the vertical fundamental], laws come ‘closer and closer’ to things, until there comes the lowermost layer, where laws are not distinguishable from things at all. We try to justify this next. We will argue that we do not see the proof in material things because these material things are treated as being distinct from space and time.

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Source: Aguirre A., Foster B., Merali Z. (Eds.). What is Fundamental? Springer,2019. — 189 p.. 2019

Is the Law a Thing?

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