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tanszek:oktatas:techcomm:information_-_basics:sciences [2024/09/12 17:53] – [Deductive Sciences] kneheztanszek:oktatas:techcomm:information_-_basics:sciences [2025/09/15 17:52] (current) – [Inductive Sciences] knehez
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 ====== What is science? ====== ====== What is science? ======
  
-According to the definition: //Science// is understood as the provable and fact-based system of the objective relationships between //nature//, //society//, and //thinking//.+According to the definition: //Science// is understood as a provable and fact-based system of the objective relationships between //nature//, //society//, and //human thought//.
  
-//Science// is not just a collection of knowledge, but a discovery process. //Science// aims to discover new information, facts, and answers about our world or the universe. +However, science is not only a set of theories stored in textbooks. It is a dynamic process of exploration and innovation that directly shapes technology, industry, and daily life. Science is the reason we can design sustainable energy systems, create AI-powered applications, or even send rockets into space.
  
-//Science// is distinguished from other historically established forms of social consciousness by the following characteristics:+//Science// is not just a collection of knowledge, but an ongoing **discovery process**. It aims to discover new information, facts, and answers about our world or the universe, while also solving real-world engineering problems.
  
-//Science// has been highlighted because of the following criteria from our historically established social forms of consciousness:+Science, among our historically established forms of social consciousness, is distinguished and emphasized by the following characteristics.
  
-  * they possess high-reaching concepts or logical tools to formulate or express broad, general or universal **principles** or **laws** (e.g. gravity, axioms, [[https://en.wikipedia.org/wiki/Maxwell%27s_equations|Maxwell's equations]])+  * It possesses high-reaching concepts or logical tools to formulate or express broad, general or universal **principles** or **laws** or **axioms** (e.g. gravity, axioms, [[https://en.wikipedia.org/wiki/Maxwell%27s_equations|Maxwell's equations]])
      
-  * they can describe the objective **conditions** under which these principles or laws will prevail.+  * It can describe the objective **conditions** under which these principles or laws will prevail.
  
-  * they possess the required logical tools or methods that can help us to calculate or predict **results** in given circumstances,+  * It provides logical methods and computational tools that enable us to calculate, simulate, and predict **results** under specific circumstances.
  
-According to **principles**, **conditions** (circumstances), and **results** (these three general aspects) we can categorize every scientific problem into the following problem groups.+From a modern engineering viewpoint, these features mean that science is not abstract—it is applied, testable, and useful. Every time an engineer uses mathematical models to optimize a production line, simulates a digital twin of a factory, or analyzes big data to predict outcomes, they are applying these scientific principles. 
 + 
 +According to the three general aspects — **principles**, **conditions**, and **results** — we can categorize every scientific problem into the following groups. 
 + 
 +----
  
 ====== Inductive Sciences ====== ====== Inductive Sciences ======
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 **Explanation:** //Induction// is probably the most important logical method used by scientists to draft new //theories// or //principles//. **Explanation:** //Induction// is probably the most important logical method used by scientists to draft new //theories// or //principles//.
  
-Induction is a generalizing method, which means that we seek a universal or general law from a given set of data with fixed conditions. A well-known example of this method is the [[https://en.wikipedia.org/wiki/Mendelian_inheritance|Mendelian laws of inheritance]].+Induction is a generalizing method, which means that we seek a universal or general law from a given set of data with fixed conditions. A well-known example of this method is the [[https://en.wikipedia.org/wiki/Mendelian_inheritance|Mendelian laws of inheritance]] or [[https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion| Kepler's law of planetary motion]]
  
-The biggest problem with this method is whether we have (or have yet to) carry out sufficient observations to arrive at a general conclusion.+The biggest problem with this method is whether we have (or have yet to) carry out //sufficient observations// to arrive at a general conclusion.
  
 In natural sciences, we are always dealing with partial induction. The more experiments we do, the more confident we will become and the better our chances of understanding the connections. In natural sciences, we are always dealing with partial induction. The more experiments we do, the more confident we will become and the better our chances of understanding the connections.
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 **Remark**: Legislative processes are based on an inductive method that analyzes social problems and their causes and makes new laws as a conclusion. **Remark**: Legislative processes are based on an inductive method that analyzes social problems and their causes and makes new laws as a conclusion.
  
-**Example**: In **information technology**, mathematical induction can be applied to many areas, including algorithm analysis and data structures. One typical example is proving the correctness of algorithms or the properties of data structures like trees. Here’s an example from **binary trees**:+---- 
 + 
 +**Example 1**: In **information technology**, mathematical induction can be applied to many areas, including algorithm analysis and data structures. One typical example is proving the correctness of algorithms or the properties of data structures like trees. Here’s an example from **binary trees**:
  
 **Problem**: **Problem**:
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   - **Inductive Hypothesis**: Assume that for any binary tree with \(k\) nodes, the number of edges is \(k-1\).   - **Inductive Hypothesis**: Assume that for any binary tree with \(k\) nodes, the number of edges is \(k-1\).
   - **Inductive Step**: We must prove that if the statement holds for a binary tree with \(k\) nodes, then it also holds for a binary tree with \(k+1\) nodes. \\ Suppose we add one more node to the binary tree, bringing the total number of nodes to \(k+1\). When we add this node, we also add exactly one edge connecting the new node to an existing node in the tree (either as a left or right child of a parent node). \\ \\ By the inductive hypothesis, the tree with \(k\) nodes has \((k - 1)\) edges. Adding one more node introduces one additional edge, so the number of edges in the tree with \((k + 1)\) nodes is: $$ (k-1) + 1 = k $$ This matches the formula for the number of edges in a tree with \((k + 1)\) nodes, which should be \((k-1) + 1 = k\).    - **Inductive Step**: We must prove that if the statement holds for a binary tree with \(k\) nodes, then it also holds for a binary tree with \(k+1\) nodes. \\ Suppose we add one more node to the binary tree, bringing the total number of nodes to \(k+1\). When we add this node, we also add exactly one edge connecting the new node to an existing node in the tree (either as a left or right child of a parent node). \\ \\ By the inductive hypothesis, the tree with \(k\) nodes has \((k - 1)\) edges. Adding one more node introduces one additional edge, so the number of edges in the tree with \((k + 1)\) nodes is: $$ (k-1) + 1 = k $$ This matches the formula for the number of edges in a tree with \((k + 1)\) nodes, which should be \((k-1) + 1 = k\). 
 +
 +----
 +
 +**Example 2**: Sum of consecutive natural numbers. **Claim**:
 +
 +$$ 1 + 2 + \cdots + n = \frac{n(n+1)}{2}. $$
 +
 +Base case (n=1):
 +
 +$$ 1 = \frac{1 \cdot 2}{2} = 1, $$
 +
 +so the statement holds.
 +
 +**Inductive step**:
 +
 +Assume the formula is true for \(n = k\):
 +
 +$$ 1 + 2 + \cdots + k = \frac{k(k+1)}{2}. $$
 +
 +Then for \(n = k+1\):
 +
 +$$ 1 + 2 + \cdots + k + (k+1) = \frac{k(k+1)}{2} + (k+1). $$
 +
 +**Simplify**:
 +
 +$$ \frac{k(k+1)}{2} + (k+1) = \frac{k(k+1) + 2(k+1)}{2} = \frac{(k+1)(k+2)}{2}. $$
 +
 +Thus, the formula also holds for \( n = k+1 \). Thus, by mathematical induction, the formula holds for all \(n\in\mathbb{N}\)
 +
  
 https://en.wikipedia.org/wiki/Mathematical_induction https://en.wikipedia.org/wiki/Mathematical_induction
 +
 +----
  
 ====== Deductive Sciences ====== ====== Deductive Sciences ======
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 The quote from [[https://en.wikipedia.org/wiki/Niels_Bohr|Niels Bohr]], //"In the Institute, we only have one piece of experimental equipment: a ping-pong table"// is a good example of reductive reasoning in science. It suggests that groundbreaking discoveries can be achieved not through extensive experimental setups but rather through the adjustment of theoretical laws and conditions. Bohr highlights the power of thought experiments and abstract reasoning, emphasising that manipulating underlying principles can lead to new insights without always needing physical experimentation. The quote from [[https://en.wikipedia.org/wiki/Niels_Bohr|Niels Bohr]], //"In the Institute, we only have one piece of experimental equipment: a ping-pong table"// is a good example of reductive reasoning in science. It suggests that groundbreaking discoveries can be achieved not through extensive experimental setups but rather through the adjustment of theoretical laws and conditions. Bohr highlights the power of thought experiments and abstract reasoning, emphasising that manipulating underlying principles can lead to new insights without always needing physical experimentation.
 +
 +----
  
 ====== Reductive Sciences ====== ====== Reductive Sciences ======
tanszek/oktatas/techcomm/information_-_basics/sciences.1726163606.txt.gz · Last modified: 2024/09/12 17:53 by knehez