Nature of Mathematics

Module Overview

• This module is for NSCI 110: Science, Technology, and Society, taught at West Visayas State University.
• Richelle O. Tuvillo is the team leader/coordinator.
• Larry D. Buban is the team editor.
• Authors and contributors include Larry D. Buban, Harlan C. Dureza, Eileen L. Loreno, Grace A. Manajero, Anita Estela M. Monroy, Vivien Mei C. Reyes, Stephen G. Sabinay, and Agatha Z. Senina.
• The module is for the College of Arts and Sciences, Physical Science Department.

Introduction to Unit 1

• Unit 1 is an Introduction to Science, Technology, and Society.
• It correlates to NSCI 110.
• Harlan C. Dureza is the author of Unit 1.
• There are six lessons in Unit 1.
• Lesson 1 focuses on the Nature of Science, including the scientific world view, methods of inquiry, and the scientific enterprise.
• Lesson 2 focuses on the Nature of Mathematics, covering patterns, relationships, and mathematical inquiry in science and technology.
• Lesson 3 focuses on the Nature of Technology, addressing technology's relation to science, designs and systems, and issues in technology.
• Lesson 4 focuses on The Physical Setting, including the universe, the Earth, the structure of matter, energy transformations, motion, and forces of nature.
• Lesson 5 focuses on The Living Environment, including diversity of life, heredity, cells, interdependence, flow of matter and energy, and evolution.
• Lesson 6 focuses on The Human Organism, including human identity and development, basic functions, learning, and both physical and mental health.

Lesson 1: Nature of Science

• The lesson explores interconnected, validated ideas about the physical, biological, psychological, and social worlds developed over human history.
• Science, mathematics, and technology form the scientific endeavor.
• The lesson focuses on the scientific world view, scientific methods of inquiry, and the nature of the scientific enterprise.

Lesson 1: Learning Outcomes

• Students will be able to explain how science works and what science has exactly explained.
• Students will discuss where science begins and ends.
• Students will be able to explain the development of many interconnected and validated ideas about the physical, biological, psychological, and social worlds.
• Students will understand the means to develop observations, thoughts, experiments, and validation, which represent the nature of science and its difference from other modes of knowing.

Galileo's Story

• Galileo was taught that the Earth is the center of the universe.
• Galileo observed the moon with his telescope, discovering mountains and valleys.
• His observations supported the belief that other worlds might be similar to earth.
• Galileo observed the moons of Jupiter, challenging the idea that all celestial bodies revolved around the Earth.
• While Johannes Kepler supported Galileo's ideas, Galileo used a secret code to share discoveries with Kepler, to prevent Kepler from claiming the ideas as his own.
• Galileo's work supported Copernicus' heliocentric ideas.
• The Church ordered Galileo not to teach Copernicus’ theories, then decided to alter Copernicus' book to protect faith in the Bible.
• Galileo was convicted for disobeying the Church's instructions.
• He agreed to publicly apologize and was placed under house arrest, during when he continued to work on science ideas.

Analyzing the Scientific World View

• The Scientific worldview shows scientists share basic beliefs and attitudes about their work regarding the nature of the world.
• Science presumes events in the universe occur in consistent patterns understandable through study.
• Scientists believe intellect and instruments can discover nature's patterns.
• Science assumes a single system where knowledge gained in one part applies universally, citing motion and gravitation.

Changing Scientific Ideas

• Science utilizes observations and theories to produce knowledge.
• New observations may challenge existing theories.
• Scientists assume increasingly accurate approximations can account for the world, acknowledging no complete truth is attainable.

Durable Scientific Knowledge

• Scientific knowledge evolves through idea modification rather than outright rejection.
• Einstein's theory of relativity incorporates rather than discards Newtonian motion laws.
• Scientists' ability to make precise natural phenomena predictions indicates growing understanding.
• Continuity and stability characterize science along with change and tentativeness.

Science's Limitations

• Science is unable to examine anything that cannot be proved or disproved.
• Science lacks the means to settle issues concerning good and evil.

Scientific Inquiry

• Scientific disciplines rely on evidence, hypotheses, theories, and logic.
• Scientists differ in investigation phenomena, data reliance, methods, and recourse to principles.
• There is a regular exchange of information, techniques, and concepts across scientific circles.

Evidence and Science

• Validity of claims is settled by observations of phenomena.
• Scientists focus on accurate data from settings ranging from natural to contrived.
• Instruments enhance senses.
• Scientists take observations passively, make collections, and/or actively probe.
• Scientists may control one condition at a time to identify its exclusive effects.
• Great value is placed on better instruments and checked group findings.

Logic, Imagination, and Science

• Scientific arguments must conform to logical reasoning principles.
• Scientists formulate hypotheses to guide data attention and interpretation.
• A hypothesis suggests what evidence would support/refute it.
• Scientific concepts don't emerge automatically from data or analysis alone.
• Knowledge and insights are required to recognize meanings.

Explaining and Predicting in Science

• Scientists explain phenomena using existing scientific principles.
• Scientific theories must be logically sound and supported by observations.
• Credibility of scientific theories comes from showing relationships among unrelated phenomena.
• Demonstration of predictive power does not need future events, can be about past evidence.
• Theories are tested by new discoveries.

Identifying and Avoiding Bias

• Scientists ask what evidence supports a claim,.
• Scientific evidence can be biased if interpretation, data handling, or data selection is flawed.
• Scientists’ background may incline emphasis of evidence types or interpretations.
• Bias by investigators, samples, methods,/instruments is difficult to avoid.

Science Without Authoritarianism

• Knowledgeable sources are turned for information and opinions.
• Authorities have been wrong many times in history.
• No scientist is empowered to decide for others what is true.
• New ideas not meshing with mainstream may face vigorous criticism.
• Theories are judged by results.

The Scientific Enterprise

• Science has individual, social, and institutional aspects.
• Scientific activity distinguishes contemporary from earlier times.

Science as a Socially Complex

• Scientific work involves many people and diverse tasks worldwide.
• Participants focus on scientific knowledge for its sake or for practical purposes.
• Data gathering, theory building, and communicating are involved.
• It reflects social values/viewpoints, affecting direction influenced by science-internal culture.
• Committees regularly review disciplines process and recommend general funding priorities.

Importance Dissemination of Information

• Dissemination facilitates scientific progress and is presented through papers, informs others.
• Papers expose ideas to peer criticism as well as worldwide developments.
• The advancement of information science and tech affects all sciences, speeds data, makes analysis practical and shortens discovery.

Institutions, Disciplines, and Philosophies

• Universities, industry, and government all play scientific roles.
• Physics informs chemistry, astronomy, etc., and continuously forms new disciplines.
• Disciplines possess their own histories, studies, techniques, and desired outcomes.

Scientific Ethics

• Conduct should follow the norms, keep accurate records, be open, replicate, and have peer reviews.
• Pressure to publish first can lead to holding off from the information.
• Violating nature impedes, condemns and impacts those who fund.
• An ethical domain relates to possible experimenting harm, like the treatment of live subjects.
• Research involving human subjects needs their informed consent entailing full risk/benefit disclosure.

Public Affairs and Science

• Scientists can provide information, insights and assist representativeness and disasters.
• They try to distinguish interpretations from research findings and apply scientific principles.
• Results tend to show only matters in public debate and there remains not a broad knowledge amount scientists.
• They avoid bias, however personal ties can present stakes and this influence their thoughts.

Lesson 2: Nature of Mathematics

• Relies on logic and creativity and is pursued both for a variety of practical purposes and for its intrinsic interest.
• Its essence lies in its beauty and its intellectual challenge.
• The chief value of mathematics is how it applies to their own work.
• A basic understanding of mathematics is requisite for scientific literacy.
• Students need to perceive mathematics as part of the scientific endeavour, as an applied science.

Patterns and Relationships

• Mathematics is the science of patterns and relationships that explores possible ones between abstractions.
• Abstractions have no concern for real world connections from numbers to geometric figures to equation sets.
• Finding the greatest economy and simplicity of representation is key.
• Enables insights developed in regards to various parts.
• It also an applied science focusing on problem-solving originating from experience.

Science, Technology, and Mathematics

• Mathematics provides science with powerful tools for data analysis.
• Investigating and studying abstract patterns has proven to be very useful to science.
• Both are trying to discover general patterns/relationships and are part of same endeavor.
• The symbolic language of mathematics has turned out to be extremely valuable for expressing scientific ideas.
• Mathematics provides the grammar of science-the rules for rigorously analyzing ideas and data.
• A belief of understandable order; an interplay of imagination and rigorous logic are included.
• With further advancement, technology will open of new fields investigation.

Key Phases in Mathematics

• Using mathematics to express ideas or to solve problems involves at least three phases: (1) representing some aspects of things abstractly, (2) manipulating the abstractions by rules of logic to find new relationships between them, and (3) seeing whether the new relationships say something useful about the original things.

Abstraction in Math

• Mathematical thinking often begins with abstraction.
• Aspects have common-ground (concrete/hypothetical) is represented through symbols.
• Abstractions aren't only from concrete objects/processes; they too can be produced with numbers.

Mathematical Statements

• After statements have been created and rep, symbols can combine as requested.
• Typically, symbol strings combine in statements expressing propositions, that combine for the side the proposition depends on to make sense.
• Mathematical insights into abstract relationships have grown over thousands of years.

Lesson 3: Nature of Technology

• From the long techniques of having people using tools, it is taken as evidence to see human culture.
• On technology as well as it has linked with scientific work, there has been great force in civilizations which has shaped civilization.
• There's much on recommendation from literacy, like emphasizing the thinking towards wisdom.

Technology & Science

• Contributes to and draws on in science.
• Previously, it stemmed from people's insights now though a mix of literature it has gained contribution of science.
• Systematic application comes through developing systematically.
• Scientific knowledge serves to act as a means in which it estimates how behaviour can act.

Scientific Values in Engineering and Inquiry

• Best is that which combines inquiry.
• It combines for in broad designing and finding.
• Method for the details including automobile, mechanics, etc.

Explore the core concepts of mathematics, its correlation with science and technology. Understand its characteristics and how it fosters interconnected ideas across generations. Learn about methodologies for establishing formulas.