Are Real Numbers "Real"?

As mentioned in my previous post, the scientific method is just about all the philosophy an engineer needs to know. In this post, I describe how this attitude can be applied to all the mathematics that engineers use.

The scope of the scientific method is defined by the assumption that the sole test of knowledge is experiment. That is, that which is outside the scope of experimental confirmation or falsification is not knowledge. Thus, to count mathematics as knowledge means that its axioms, operations, and results must be experimentally confirmed.

But there are inherent limits to the precision of experimental measurements in the real world. No quantity or phenomenon can be measured with absolute precision and no process can be performed without at least an infinitesimal possibility of error. For example, there is no experiment that can exactly measure any irrational, rational, or natural number. And since these numbers claim to be exact, this means they do not exist in reality. There is no zero because absolute zero cannot be observed. Just the infinitesimal. There is no infinity for the same reason. Just the very huge.

Does this affect the mathematical techniques useful to engineers -- such as the calculus? Are these techniques to be abandoned or viewed with suspicion? Not at all. Abraham Robinson was able to incorporate infinitesimals and huge numbers into what is called nonstandard analysis. Thus, the techniques remain the same, just the nature of the proofs change.

Interestingly, there is a number that is totally incommensurate with any attempt at direct measurement or observation yet inferred by the observed laws of Nature. It is the square root of minus one. In this sense it is the only number that is purely "imaginary."

The Scientific Method

The scientific method is just about all the philosophy an engineer needs to know. The scientific method refers to a framework of techniques for acquiring, correcting, and integrating knowledge. Although there are disagreements on the practical details, the philosophy behind the basic cyclical process enjoys a large consensus. The process can be illustrated as follows:

 

Theory/Model - Prior to new experimental evidence, this can be viewed as the initial state of the cycle and represents the relevant knowledge (hypotheses, theories, models, etc.) that is to be tested. After the experiment has been performed, this can be viewed as the final state that represents the (possibly conditioned) relevant knowledge as modified by the new experimental evidence.

Deduction/Simulation - This action represents the process of exercising the logical/mathematical consequences of the theory/model in order to yield a consequence capable of being experimentally tested.

Prediction/Forecast - Based on the prior theory/model, this state represents some sort of explicit statement of what the outcomes (observations/results) of the experiment are predicted or forecast to be.

Experiment - This action represent the actual setup and performance of the experiment in order to test the predictions/forecasts.

Observations/Results - This state represents the evidence (facts/data) acquired by the experiment.

Abduction - This action represents the process (logical induction) of modifying/replacing (if necessary) the prior theory/models to make them more consistent/agreeable with the latest experimental evidence.

IV&V - These actions performed by independent stakeholders (Independent Verification and Validation) take place throughout the cycle. These quality assurance processes (such as peer review) are necessary to reduce the risk of error to an acceptable level.

The fundamental assumptions upon which the scientific method rests (that is, that which is more-or-less undefined and simply taken on faith by all stakeholders) are approximately:
  1. Theory shall be logically consistent. (For example, the interpretation of experimental evidence as Bayesian.)
  2. Theory and experiment shall be parsimonious. (For example, Occam's Razor.)
  3. The sole test of theory shall be experiment. (Feynman's 'almost' definition of science.)
  4. All experimental processes and evidence shall be independently verified and validated.
The above assumptions are necessary and sufficient for scientific objectivity within the realms that perform IV&V. I use the word 'shall' above simply to reinforce the concept that these are simply rules taken on faith.

Why these assumptions? The first assumption is necessary to promote rational discourse about science. Otherwise, consensus is unobtainable. The second rule is 'merely' practical and helps control error and makes IV&V easier. The third rule is the most fundamental one and key to any process being labeled as 'scientific'. The fourth rule is needed only to the extent that fallible beings are used to conduct science.

Note that even this most bare form of the scientific method contains two logical fallacies. The first is the use of abduction (affirming the consequent). The second is the partial reliance on IV&V for error management (appeal to authority). The use of abduction eliminates logical certainty from the scientific method and introduces the possibility of error. The logical shortcoming of IV&V means that finding and eliminating error is never certain.

Also available is a Bayesian version of this approach to the scientific method.