Science informs us of the nature of our reality. Mathematics provides us a logical metaphysical construction that offers utility to science. Scientific Method is the label we give to a common strategy used by scientists and mathematicians in their pursuit of knowledge.

Mathematics and science may seem quite distinct types of knowledge, but there is a wonderful synergy in which the metaphysical knowledge of mathematics can be used as a tool and language to extend our perceptions of reality and to further inform our intuitions.

Let’s look at the analogous ways that science and mathematics create knowledge. They share a three step process:

- observation – determine characteristics of observed phenomena related to a question
- induction – generalize/analogize from the observations to form an abstract hypothesis
- validation – verify the hypothesis, answering the question

The ways of science and mathematics differ in details of the observation and validation steps, and in the kind of knowledge that results.

The way of science is based in observations of real world phenomena and their interactions. The collected and verified scientific evidence from our senses and measurements, together with the scientist’s knowledge and experience, will lead the scientist to make a guess (plausible inference) regarding the how and why of the observed phenomena.

The way of mathematics is straightforward, even if the advanced results strain comprehension. Mathematicians abstract sets of objects and operators and observe their patterns and interactions. Like science, mathematics proceeds by induction from specific observations to a general hypothesis.

In mathematics, the observations are logically certain. Toward a similar necessity, only empirically verifiable real-world observations can qualify as inputs to the scientific method (avoiding the consequence: garbage in = garbage out).

Mathematical validation consists of a proof, an application of deductive logic, proceeding from axioms. Not all hypotheses have proofs yet. Those that don’t languish as conjectures. Scientific validation is experimental. The experiments are designed to test the scientific hypothesis. Thus only testable hypotheses are compatible with the scientific method. (See my page on Pseudo-Science, a discussion of a pretend science based on unverifiable observations and untestable hypotheses.)

Whenever any scientific experiment fails, the hypothesis is modified appropriately and subjected to further experimentation. A successful experiment will verify the hypothesis beyond reasonable doubt. The more experiments that verify the hypothesis, the stronger (more plausible) is our belief in its validity. But our belief never gets to certainty. And if any subsequent experiment fails, the hypothesis is again revisited to look for a modification to account for the failure.

We assume that rational minds pursue the scientific method, so that all evidence is evaluated dispassionately. But scientists are human beings susceptible to human frailties originating in our emotional brains. Thus, a scientist may gain an emotional vested interest in a hypothesis, and then unconsciously skew their validation data analysis in favor of the hypothesis. This insidious slanting of results is more damaging to science than can be any fake posturing by the overt science deniers and pseudo-science practitioners. All scientists need to raise an alarm to themselves whenever a hypothesis’ success assumes more importance to their emotional state than does an understanding of how things really work.

It is instructive to contrast scientific knowledge with knowledge that comes from the way of mathematics. Both types of knowledge provide a sense of certainty, but within bounds. The great success of the way of science and mathematics is measured by how little this absence of certainty restricts growth of knowledge. The construction and experimentation goes on merrily.

For mathematics, the bounds on certainty derive from the adequacy of our chosen logic system, and of the chosen axioms. Pragmatically, working within these bounds of logic and axioms is sufficient for almost all mathematicians, and for all of the rest of us. The remaining few are thinking outside these bounds in an attempt to answer how well we know what we know.

In science, the bounds on certainty are perceptual, involving imperfection of our senses, imperfect hypothesis and validation strategy, inadequacy of the sensitivity and power of instruments we build to assist them, potential inadequacies in the mathematics we create to extend our perceptions beyond these limits. Peer review of hypotheses and validation strategy can expose inadequate consideration of external interactions and boundary conditions that might result in false positives. In a recent simple case, hypotheses based on signals thought to originate from deep space came to wrong conclusions because the signals actually emanated from earth.

Our anthropocentric reference frame denies us ability to perceive readily at other scales, where velocity and dimension are maximized and/or minimized. Beyond our perceptual limits and the sensitivity of our detection tools, we have only mathematics/intuition to guide us, and these mathematics often create currently untestable scientific hypotheses.

Scientists, like mathematicians, are pragmatic. Almost all scientists are happy working within our current perceptual bounds, inventing new hypotheses, challenging existing hypotheses, following the mathematics where it leads in an attempt to extend perception. In parallel, a few others work at the limits, applying mathematical tools and giant machines in pursuit of answers to the most fundamental existential questions.

The role of determinism in our reality is such a question. It has been famously framed via a metaphysical quandary: Does God play dice? Further questioning our concepts of time and causality, we ask what it may mean for a universe to begin and end. And we seek better understandings of concepts central to our reasoning, such as truth, time, continuum, and infinity, to eliminate some of the fuzziness in our common discourse.