4th ed. Jan 2024; 3rd ed. May 2023; 2nd ed. Dec 2009; 1st ed. Sep 2004
“It is not philosophy but active experience in mathematics itself that alone can answer the question: `What is Mathematics?'” – Richard Courant & Herbert Robbins, 1941, What is Mathematics?, Oxford University Press)
“An adequate presentation of any science cannot consist of detailed information alone, however extensive. It must also provide a proper view of the essential nature of the science as a whole.” – Aleksandrov, 1956, Mathematics: Its Content, Methods, and Meaning
‘What is mathematics?’ Much ink has been spilled over this question, as can be seen from the selection of ten respected responses provided in the footnote, with seven book-length answers, and three written in the current millenium. One might well ask, is there anything new that can be said, that should be said? We’ll start by clarifying what a good answer should look like, and then explore the answer proposed.
The rest of the paper follows the structure below:
1. Criteria for a Good Definition of Mathematics
2. Definition 1: covering mathematics up to the end of the 18th century (1790s)
3. Two Perspectives
Mathematics as Dialectic (Lakatos)
Mathematics shaped by its Anthropology (Hoyrup)
4. Definition 2: covering all mathematics, including contemporary mathematics
5. The emergence of contemporary mathematical practice from 1800s onward
6. Three Facets of Mathematics
1. Mathematics as an Empirical Science
2. Mathematics as a Modeling Art
3. Mathematics as an Axiomatic Arrangement of Knowledge
7. Mathematics "from the inside": Mathematicians writing about Mathematics
8. Continue Reading
9. References
Continue reading this article…
*New!* (29 Aug 2020) – Turtle Logo v1.8 (portable) is available! Developer kit with source code included. Suitable from ages 3 years to adult. (970 lines of Forth code).
1. Inspiring the next generation of technology builders.
A challenge facing parents and teachers is how to help children develop ‘builder’ relationships with technology rather than being limited to the passive consumption of content created by others. The consensus on what’s important for older kids and adults is clear: coding. This enables children to participate in the creation of their own technological “micro-worlds” — environments rich in educational potential.[14]
This autumn, spurred by having our own young children (one aged 4 years, the other 16 months), we began an experiment, the result of which is a Turtle Logo program for Windows computers (freely downloadable) that is simple enough to be accessible for children from 3 years and older, while providing an extensible platform that can grow with the child.
The long-term goal is to enable children to express their creativity, artistry, and natural ‘builder’ impulses using coding, computer graphics, and robotics as readily as the previous generation could using paints, brushes, and building blocks.
 Turtle Logo – Inspiring the next generation of technology builders. Continue reading this article…
By Assad Ebrahim, on July 13th, 2016 (10,095 views) |
Topic: Education
Thoughts on Teaching Mathematics in an Exploratory, Dialectical, Topical format.
(2nd ed. July 13th 2016; 1st ed. Jan 31, 2010)
Mathematics is a richly spun tapestry, threaded with interconnections from a multiplicity of endeavors, perspectives, and disciplines, both theoretical and applied. Yet contrary to this “non-linear” reality, the typical pattern of school and even university mathematics is both linear and restricted.
This article takes a look at what lies behind the way things are, and what could bring positive change.
Continue reading this article…
By Assad Ebrahim, on January 2nd, 2010 (105,423 views) |
Topic: Maths--Philosophy
What are the characteristics of mathematics, especially contemporary mathematics?
I’ll consider five groups of characteristics:
- Applicability and Effectiveness,
- Abstraction and Generality,
- Simplicity,
- Logical Derivation, Axiomatic Arrangement,
- Precision, Correctness, Evolution through Dialectic…
Continue reading this article…
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