发现
Software 
See: Math, 20110402 Math Deep Structure, 20160620 Discovery in Mathematics, Math Discovery Examples Discovery in Mathematics: A System of Deep Structure I'm now rewriting the above article in two parts:
Notes
ZermeloFraenkel axioms of set theory
Also:
Implicit math: Sets are simplexes. Are simplexes defined by their subsimplexes as well?
Eightfold way
Reorganizings
Relate to multiplication Notes Total order is the same as a labeled simplex. Extension: 3! + (4 + 4 + 4 + 6) = 4! 3! = 2! + 4 (representations: 2 for edge and 4 for vertex) We may assign the weight q^(k1) to the kth vertex and the weights 1/q to each new edge. These weights give each vertex a unique label. The weight of each ksimplex is then the products of the weights of their vertices and edges. The Gaussian binomial coefficients count these weighted ksimplexes. Without the weights the vertices are distinct but there is no way to distinguish them. The symmetry group is the Symmetric group which relabels the vertices. Matematikos išsiaiškinimo būdai Ways of figuring things out in mathematics Discovery. What are the ways of figuring things out in mathematics? We can study mathematics as an activity by which we create and solve mathematical problems. The techniques and structures that we use in our minds are much more elemental than the mathematical output which they generate. I have described and systematized 24 ways of figuring things out in mathematics. I now want to relate that to an overall methodology for answering mathematical questions. A Conversation with Gary Klein. How firefighters deal with challenges onthefly. I'm giving a talk on the ways of figuring things out in Mathematics. It's based on a letter that I wrote about a comprehensive system that I organized of 24 such ways. It needs to be 10 minutes long, 1,000 words. Discovery in Mathematics: A System of Deep Structure I will talk about how we may systematically study the ways of figuring things out in mathematics. George Polya, in his book, "How to Solve It", considers Euclid's problem of how to construct an equilateral triangle. If we are given the side AB, how do we construct the other two? The solution is a recurring idea which Polya calls the "pattern of two loci". We think of there being two separate conditions. One side must extend a length AB from the point A. Another side must extend a length AB from the point B. We thus draw two circles of radius AB centered at A and B. The points where the two circles intersect are those where we can draw a third point C which satisfies both conditions so that our triagle is equilateral. I realized that our minds solve this problem by imagining a powerset lattice of conditions. Circle A is one condition, circle B is another condition, and the intersection of A and B satisifies the union of these two conditions. Our minds have thus solved the surface problem (constructing a triangle) by considering a simpler, deeper structure (a lattice of conditions). This brings to mind linguist Noah Chomsky's work in syntax and architect Christopher Alexander's work on pattern languages. I collected such problem solving patterns discussed in Paul Zeitz's book "The Art and Craft of Problem Solving" and other sources. Each distinct pattern makes use of a structure which is familiar to mathematicians and yet is not explicit but mental. We may consider those math structures to be cognitively "natural" which are used by the mind in solving math problems. I present to you 24 patterns which I identified and systematized them in a way which suggests they are complete. The system distinguishes between We may always start a fresh sheet (independent trials). cognitive structures used on a single mental "sheet", as in algebra (center, balance, polynomials, vector spaces), those that suppose an endless sequence of sheets, as in analysis (sequence, poset with maximal or minimal element, least upper bounds or greatest lower bounds, limits). Sheets may be "stitched together" (extend the domain, continuity, selfsuperimposed sequence). Algebraic and analytic approaches thus combine to give a complete, explicit system (symmetry group). Within a system, we can relate an explicit sheet where it has fixed expression with an implicit mental sheet where it is imagined and worked on (truth, model, implication, variable). We can then manipulate a problem explicitly as one of six visualizations of a graph structure (tree of variations, adjacency graph, total order, powerset lattice, decomposition, directed graph). However, the meaning of any explicit expression such as 10+4= ultimately depends on what is implicit, whether we have in mind, for example, the integer Z or a clock Z12 (context).
George Polya's book "How to Solve It" showed how we can collect recurring cognitive "patterns" which mathematicians apply intuitively in a variety of settings. For example, in drawing an equilateral triangle, given the first side AB, how do we draw the other two? Using the "pattern of two loci", we draw circles of radius AB centered at A and B and find their intersections. I note here that, in solving this problem, our mind constructs a lattice of conditions on the solution: the plane (no condition), circle A (one condition), circle B (one condition) and the intersection of A and B (two conditions). Thus the surface problem (constructing a triangle) is solved in the mind by considering a simpler, deeper structure (a lattice of conditions). This brings to mind linguist Noah Chomsky's work in syntax and architect Christopher Alexander's work on pattern languages. I collected such problem solving patterns discussed in Paul Zeitz's book "The Art and Craft of Problem Solving" and other sources. [1] Each distinct pattern makes use of a structure which is familiar to mathematicians and yet is not explicit but mental. We may consider those math structures to be cognitively "natural" which are used by the mind in solving math problems. I identified 24 patterns and systematized them in a way which suggests they are complete. I will present this system as drawn and described in more detail at [2]. The system distinguishes between cognitive structures used on a single mental "sheet", as in algebra (center, balance, polynomials, vector spaces), and those that suppose an endless sequence of sheets, as in analysis (sequence, poset with maximal or minimal element, least upper bounds or greatest lower bounds, limits). We may always start a fresh sheet (independent trials). Sheets may be "stitched together" (extend the domain, continuity, selfsuperimposed sequence). Algebraic and analytic approaches thus combine to give a complete, explicit system (symmetry group). Within a system, we can relate an explicit sheet where it has fixed expression with an implicit mental sheet where it is imagined and worked on (truth, model, implication, variable). We can then manipulate a problem explicitly as one of six visualizations of a graph structure (tree of variations, adjacency graph, total order, powerset lattice, decomposition, directed graph). However, the meaning of any explicit expression such as 10+4= ultimately depends on what is implicit, whether we have in mind, for example, the integer Z or a clock Z12 (context). Notes
(polynomial coefficient) Ordinal/List/Analysis vs. Cardinal/Set/Algebra (polynomial root)  the coefficients and roots are related by the binomial theorem, the factors are choices.
Think of monads as describing models, that is, simplified accounts of a system. Note that monads are built on adjoint functors, which express least upper bounds and greatest lower bounds. So this suggests that the four levels of logic/geometry are related to the four levels of algebra/analysis. But how, perhaps inversely? Because the bounds are third level, but the models are second level. Thus:
What this suggests is that the richest structures are inherently contradictory. But if we pull back to simpler structures then they become safer.
