Motivation:

Most scientists are not familiar with the Church-Turing thesis, and yet as I shall show, its validity is a fundamental axiom of the scientific method. In order to clarify that human scientists are within the Turing limit, i.e. compute functions that may be computed by Turing Machines, I shall present five different arguments on the matter.

Animal brains have finite memory:

All the neurobiological evidence suggests that mammals have a finite number of sensory and behavioural states. In the worst case, we may apply the Bekenstein bound to the human sensorimotor system and deduce that the human brain has finite memory.

The activity of animal brains may be simulated by Turing Machines:

The application of the Second Law of Thermodynamics to homeostatic regulation places a constraint on the directionality of information processing in the human brain. This means that processes may be grouped into sequential operations such that they satisfy the Principle of Minimum Energy. This observation and the all-or-nothing operation of neurons suggests that the human brain may be simulated by a Turing Machine.

The limits of our language are the limits of our world:

All physical systems at present are modelled within a mathematical framework that is consistent with Peano Arithmetic(PA) for combinatorics and algorithm design, first order logic for mathematical reasoning and proof verification, and a theory of types that is implicitly necessary for physical equations to make sense. This means that human scientists and the models they define operate within the confines of the Church-Turing thesis.

On the other hand, if physicists reasoned within a mathematical framework that was inconsistent with PA then scientific induction would no longer be possible. So they might develop theories but they would not be testable.

It follows that if computational neuroscientists build a testable model of the brain, this model must be within the Turing limit. In a very precise sense the limits of our language define the limits of our world, but the language of mathematics is also unmatched in its expressive power.

The power of the mathematical language is unmatched:

Wigner’s empirical observation of the immense progress of the mathematical sciences in the last four hundred years suggests that for every physical system there is a corresponding computational model that may be simulated by a Turing Machine [8]. How might we explain this ‘unreasonable’ effectiveness of mathematics?

The most plausible explanation comes from theoretical computer science, the Physical Church-Turing thesis [9]. This thesis may be broken into two separate claims:

  1. Every computational process is ultimately a mathematical description of a physical process in terms of (possibly coarse-grained) state transitions.

  2. Every physical process has a mathematical description in terms of a sequence of mathematical operations such that it may be simulated by a Turing Machine. Therefore, every physical process may be simulated by a universal computing device.

The argument of non-computable functions:

The fact that there are non-computable functions that scientists/engineers would like to compute, and can’t, strongly indicates that as far as science is concerned the human brain is within the Turing limit.

All scientific models that make predictions are computable:

Given a dataset of scientific observations, any computational model of the observed data which a human can come up with may be well-approximated by a machine learning model(ex. deep neural network) that may be simulated by a Turing machine.

Discussion:

By this line of argument, I don’t mean to say that hypercomputation beyond the Turing limit is impossible. Indeed, it appears that this might be possible in the near future once we develop the ability to engineer black holes [1]. However, the physical processes that are possible within a black hole are qualitatively different from those that are possible within the human brain.

I believe that one day we may be able to transcend the Turing limit, but for that we will need to master the principles of blackhole engineering.

References:

  1. Hajnal Andréka et al. Closed Timelike Curves in Relativistic Computation. 2011.
  2. Turing, A., 1950, “Computing Machinery and Intelligence,” Mind, 59 (236): 433–60.
  3. G. Pólya. Mathematics and Plausible Reasoning. 1954.
  4. Eric R. Kandel, James H. Schwartz, Thomas M. Jessell, Steven A. Siegelbaum and A.J. Hudspeth. Principles of Neural science. Elsevier. 2012.
  5. Dyson & Abbott. Theoretical Neuroscience. The MIT Press. 2005.
  6. Nils Nilsson. The Quest for Artificial Intelligence. Cambridge University Press. 2013.
  7. Jacob D. Bekenstein (2008) Bekenstein bound. Scholarpedia, 3(10):7374.
  8. Eugene Wigner. The Unreasonable Effectiveness of Mathematics in the Natural Sciences. 1960.
  9. Aidan Rocke (https://cstheory.stackexchange.com/users/47594/aidan-rocke), Understanding the Physical Church-Turing thesis and its implications, URL (version: 2021-02-22): https://cstheory.stackexchange.com/q/48450