Is math a hard science
Science and dogma
of Arnold Neumaier
Dogma is often considered a characteristic of religions such as Christianity. But the natural sciences need dogmas just as much as the religions in order to be communicable over a wide area.
Dogma is the codified conception of reality in a delimited area of human culture. The correspondence with reality is assumed - regardless of the extent to which it is given.
The dogma is the base on which the rest are built. If one undermines the dogma, the structure fluctuates over it, and communication becomes ambiguous, dubious. Hence the dogma is taught in schools and universities.
On the other hand, the trustworthiness of a dogma, and thus its cultural value, depends on how well it reflects the intended aspects of reality. And that depends on how easily the personal experience of the people who are supposed to share the dogma can be reconciled with it. That is why schools and universities also teach how the trustworthiness of dogmas comes about. The latter, however, is an act of advertising, because what is trustworthy for someone is determined by that person, and nobody for him.
Mathematics is limited to statements that exist between formally defined concepts and therefore has the highest degree of verifiability. One only needs to check that the chain of evidence is logically convincing. The advertisement for the dogmas of mathematics is therefore: Everything becomes transparent; everything can be strictly proven. You don't need any experiments or authorities. And it is useful too (even if the layperson is seldom aware of the extent to which modern everyday life is saturated with mathematics).
However, authorities and their values definitely play a role in the establishment of the mathematical structures of thought - because correctness is only the minimum requirement for mathematics; quality comes from choosing the most useful truths from the pool of all possible such. And what is useful depends on the point of view and is subjective. Some love beauty, others love generality, others still use applicability, and what exactly is meant by these criteria also changes from mathematician to mathematician.
Physics and chemistry are limited to statements about the materials in the world that are the same everywhere and can be reproduced at will. This also requires a high degree of verifiability, at least in everyday phenomena that are easily reproducible. The advertisement for the dogmas of physics and chemistry is therefore: Everything has been proven experimentally and, with sufficient effort, reproducible. And it is also useful, as the many technical applications and achievements show.
It is difficult, however, to check experiments that are so complex that they are rarely done. Hardly anyone quickly builds a large particle accelerator to test whether the hadron spectrum is correctly specified in the textbook. The majority of people are therefore dependent on believing in trustworthy reporting. Even the best experts can only verify a tiny fraction of our culture's knowledge of physics and chemistry firsthand. Therefore, authorities and their values (especially with regard to their truthfulness) play a much greater role in the establishment of the physical and chemical structures of thought. But they are under the control of - sometimes many, sometimes few - colleagues who are able to independently verify their statements.
The average physicist or chemist with a doctorate in chemistry learns, in addition to the content-related dogma, a canon (also dogmatically prescribed) for correct scientific behavior, which requires a certain degree of self-criticism and care in the interpretation of raw facts and read information, and thus offers a certain protection against jumping to conclusions . However, the typical graduate has only demonstrated in a few dozen experiments that the dogmas conveyed in the textbooks have an experimentally reliable basis. Nevertheless, they should believe that the whole thing is reliable and that the authorities can be trusted. (Worse still: If the results of the laboratory experiment do not agree with the textbook, then (with good reason) it is not the dogma that is rejected, but the experiment! The student is forced to bring the experiments into sufficient agreement with the dogma when he completes his internship want to complete successfully.)
In most other sciences (with the exception of parts of some sciences such as biology, medicine, and pharmacy) the knowledge is even less hard because it is less easily verifiable. Historical facts in cosmology, geology, biology, archeology and cultural history can only be verified on the basis of circumstantial evidence or belief in tradition; in most cases it is not far from reproducibility. And in cultural studies even the definitions of the terms are not undisputed, so that it is not even clear whether one is talking about the same thing when one uses the same words.
Therefore, the criteria for truth are increasingly vague and the number of competing schools with different and often incompatible explanatory models is increasing. The advertisement for the dogmas of these disciplines is therefore: Everything is scientifically proven; there is the peer review, the appraisal by competent colleagues, which ensures objectivity. But one is careful not to define the concept of scientific nature too sharply, on the one hand to benefit from the reputation of the hard natural sciences and, on the other hand, not to be drawn from the unscientific nature that is so frowned upon today.
And dogma also has its place outside the sciences; only there the criteria for truth are even more controversial, the number of schools even greater, and the precise content of the terms even more controversial. World views and religions compete for the power to determine or at least to be able to shape the ideas that people make of reality. But when an atheist, a Hindu, and a Christian speak of God, the content is almost complementary. The dogmas are mutually exclusive. In the absence of consensus in dogma, communication is difficult and talking past one another is almost inevitable.
But as long as the same dogma is accepted (or one thinks into a foreign dogma), communication is just as unproblematic and an exchange of truth is just as possible as in mathematics, physics and chemistry, where a single dogma has prevailed as the unrivaled best. But of course there are areas that are more unfamiliar to everyone, where communication becomes more difficult.
Anyone who wanted to make the world of mathematics the standard for reality would have to demand that only that which has a strictly logical proof should be recognized as the truth about reality. If taken to the extreme (i.e., taken very seriously), anything else would not be truth, as it cannot be reliably proven. Of course, that is a pointless claim, which is therefore not supported.
Anyone who wanted to make the world of physics and chemistry the standard for reality would have to demand that only what can be reproduced at will be recognized as truth. If taken to the extreme (i.e., taken very seriously), anything else would not be truth, as it cannot be reliably reproduced. Of course that is just as pointless a claim; and it is seldom represented in this sharpness.
Whoever wanted to make the world of science the yardstick for reality would have to demand that only what is scientifically proven is to be recognized as truth. If taken to the extreme (i.e., taken very seriously), anything else would not be truth because it is not scientifically proven. Of course, that is just as pointless a claim; nevertheless it is represented all too often.
However, the reality is much richer than the methodology of a discipline allows to grasp. Truth eludes a systematic verifiability secured against error on all levels of reality. Each methodology improves the understanding of what corresponds to this methodology, but creates blinkers for what eludes it.
To want to make the methodology of the sciences the standard for reality means to reduce reality to the scientific. Scientific criteria, however, do not match the unique, the unrepeatable, the personal; not on values and goals, beauty and importance. Yet there is truth and reality there. Yes, that's where the crucial thing in people's lives takes place, without dogma.
The dogma is necessary, however, in order to communicate with one another without having to clarify each time which frame of interpretation should apply.
Thoughts on living as a Christian
My Views on the Christian Way of Life
Science and Faith
my home page (http://www.mat.univie.ac.at/~neum) Arnold Neumaier ([email protected])
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