Do you know any other "language" that is used by people except mathematics and is not subject of interpretation? By subject of interpratation I mean e.g. that 1 000 000 people will undertand that 1 + 1 = 2 and never 3. E.g. when you say that "This flower is red". Some people will think that it is darker, some lighter etc.
migrated from mathematica.stackexchange.com Nov 23 '12 at 13:32
In this answer, I will address the question: are mathematicians certain of everything they say?
In retrospect I do not think this is the question that was asked (I seem to be skilled in answering the wrong questions), but I'll leave it in place nonetheless.
This is of course not possible. The absolute truth is some platonic ideal we are all reaching for. There is nothing mathematicians are really sure of, but for many things it's so very hard to doubt it, that everyone agrees - indeed, proofs for $1+1=2$ (this doesn't mean what you think it does: $2$ is defined as $1+1$, so we have to prove something else) are so very hard to debate that we agree that it is right.
And in fact, for many proofs (mathematics is all about proving things) we can systematically expand it in simpler terms, and even have a computer check them. Sure, it is possible that everyone along the line made a mistake, so that everyone thinks a proof is right but it really isn't, but this chance is so small that we say it is proven.
There have been fake proofs in mathematics. Many mathematicians claimed to have found a proof or counterexample for $P\stackrel?=NP$. Most of these "proofs" are relatively easy to refute. And in any case, unlike in the physical sciences, all data is in the proof itself, so there is nothing else you need to check the validity of an argument. In physics and the social sciences, the huge datasets that papers are based on are usually not published, and that makes it hard to check validity.
Mathematics is as much a debatable science as physics and medicine. However, we always have a hope of answering critical questions, and in this sense mathematics is "sure" of things. But in the end, even mathematics is about convincing other mathematicians that you are right.
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First, mathematics notation is subject to interpretation. Looking at your example, $1+1=0$ in modulo-$2$ arithmetic. Most mathematics requires context to be understood.
I would say that programming languages are less subject to interpretation than math; after all, machines can understand them and produce consistent results.
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During the intuitive creative process, mathematics is subject to interpretation. For example, during the invention of calculus, there were various intuitive interpretations of the notion of infinitesimals. These notions could not be made completely rigorous at the time (primarily due to lack of a precise formal language such as first-order logic). However, by and large, they produced correct results when employed by competent practitioners. Only a few centuries later was this intuition faithfully rigorized, when Abraham Robinson discovered a rigorous interpretation of infinitesimals in his nonstandard analysis.
For a more recent example, see the Wikipedia page on the notion of a field with one element, which begins "in mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist".
What is room for interpretation after all?
Statements are always subject to interpretation. To make sure there's no ambiguity in a statement you need to build a more or less complex system of axioms and corollaries that prove your statement true. The same can be applied to the flower you point at. If you state precisely which aspect of the flower you are talking about and what measure of color you apply then you can disambiguate such statement just in the same way.
While it might not always be feasible to disambiguate anything it is not always necessary or desired either. It is interesting that you came up with the example of constructivism. Indeed every person has different pictures in mind that are used to describe their sensory perceptions conceptually. This diversity is usually not a problem as there are a lot of norms and conventions "socialized" into each individual that disambiguate each others actions (including verbal utterings). Admittedly these are somewhat implicit and vary along the time and across subcultures but they provide the solid ground without which a society could not function.
As someone has mentioned computer programming to be free from interpretation, I would argue that. The programmer carefully makes explicit decisions about features and functionality, edge cases and error handling. But he usually starts with a problem statement, which he interprets. He then surrenders his code to some other application which again interprets it, and finally bring it to execution. I would argue that the average programmer knows all of the default settings that are implicitly being used by the compiler, nor is he aware of the implications of all its "optimizations". That is, the same dependency on mutually agreed conventions as in mathematics and in any other field here.
As for your example of $1+1=2$ there probably are millions of people that interpret this equation as you would expect. Such fact might be commonly accepted without any doubt, but as this question shows this is only true if the prerequisites are indeed defined accordingly. Therefore, a mathematician when being asked about the validity of $1+1=2$ should reply a fuzzy it depends, just as the attorney when being asked for legal advice would.
Only by assuming ("pretending"?) that truth and meaning reside entirely outside the human mind can one assert that "mathematics is not subject [to] interpretation". Such a view reflects a remarkably useful working consensus, broadly embraced by professional mathematicians, that, if one puts forth a set of axioms, definitions and agreed-upon rules of inference, then some propositions will be inescapably true, others will be inescapably false, and still others will be undecidable.
This point of view will often appear inconsistant with a view based on the assumption that truth and meaning are products of living, breathing individuals. In the latter view, it becomes reasonable to ask:
-how does person A interpret this set of mathematical ideas and propositions?
-how do students come to understand the basic ideas and representations of algebra?
-how can one help individuals grasp the distinction between mathematical objects (the real line, a triangle, a function) and the tokens we employ to represent them (a number line drawn on paper, a picture of a triangle, notation for an infinite set of ordered pairs)?
The mathematics educator has to be able to shift constantly between these two perspectives. In addition, s/he has to help students become adept at the former approach while remaining aware of the latter.