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tanszek:oktatas:techcomm:information_-_basics:sciences [2024/09/09 21:02] – [Inductive Sciences] kneheztanszek:oktatas:techcomm:information_-_basics:sciences [2026/09/21 12:25] (current) – [Reductive 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, Maxwell's equations)+  * It uses high-level concepts and logical tools to formulate or express broad, general, or universal **principles**, 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 take effect.
  
-  * 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. 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. 
 + 
 +Using 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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 Logic can only state that the results will be true if the premises are true (and consistent) and the arguments are logically correct. Logic can only state that the results will be true if the premises are true (and consistent) and the arguments are logically correct.
  
-//Bonus Content//:+**Example**:
  
-János Bólyai – a famous Hungarian mathematician – wrote this famous sentence to his father:+János Bólyai (1802–1860) – a famous Hungarian mathematician – wrote this famous sentence to his father:
  
 ’//I have created a new and different world from scratch//.’ He reached the conclusion that by changing the Fifth Axiom of the [[https://en.wikipedia.org/wiki/Euclidean_geometry|Euclidean principles]], a new world could be created. ’//I have created a new and different world from scratch//.’ He reached the conclusion that by changing the Fifth Axiom of the [[https://en.wikipedia.org/wiki/Euclidean_geometry|Euclidean principles]], a new world could be created.
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 {{:tanszek:oktatas:techcomm:information_-_basics:pasted:20240908-180605.png?320x220}} {{:tanszek:oktatas:techcomm:information_-_basics:pasted:20240908-180605.png?320x220}}
 +{{:tanszek:oktatas:techcomm:information_-_basics:sphere_bolyai.jpg?280|}}
 +
 +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 ======
  
-The main principles and the results are known and we are seeking the appropriate conditions which can realize our goals.+The main principles and the results are known and we are seeking the **appropriate conditions** which can realize our goals.
  
  
-**Explanation**: these type of tasks are typical examples of technical sciences.  However, sadly the solution cannot be inverted from the end results, therefore there can be an infinite number of terms which can get us to the known results. In this case we have to accept a few  possibilities (or more usually only one). We usually get to this term in heuristic ways.+**Explanation**: These types of tasks are typical examples of technical sciences. Unfortunately, the solution cannot be inverted from the end results; therefore, there can be an infinite number of terms that can get us to the known results. In this case, we have to accept a few possibilities (or, more usually, only one). We usually get to this term in **[[https://en.wikipedia.org/wiki/Heuristic|heuristic ways]]**. 
  
-We can face another interpretation of reduction in the classification of elementary scientific problems (the so-called ’Trinity’ of sciences).+We can face another interpretation of reduction in classifying elementary scientific problems (the so-called ’Trinity’ of sciences).
  
  
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   - **Main Principles Known:**   - **Main Principles Known:**
-    - The query must retrieve specific data based on given conditions (e.g., filtering, joining tables, sorting). +    - The query must retrieve specific data, based on given conditions (e.g., filtering, joining tables, sorting). 
-    - The performance depends on factors like indexing, table size, query structure, and features of database engine.+    - The performance depends on factors like indexing, table size, query structure, and features of the database engine.
     - The result of the query must remain the same regardless of the optimization.     - The result of the query must remain the same regardless of the optimization.
-  - **Seeking Appropriate Conditions:** +  - **Seeking Appropriate Conditions:** The task is to optimize the query by structuring it to minimize response time and resource usage. 
-    - The task is to optimize the query by structuring it in a way that minimizes response time and resource usage. +    - **There’s no single "perfect" solution**  for optimizing the query, because different query structures can yield the same result, but with varying degrees of performance based on the specific context (e.g., hardware, data distribution, and load). 
-    - **There’s no single “perfect” solution**  for optimizing the query, because different query structures can yield the same result, but with varying degrees of performance based on the specific context (e.g., hardware, data distribution, and load). +    - Additionally, the same query might perform differently on different database engines (e.g., MySQL vs. PostgreSQL), and therefore __there can be an infinite number of ways to structure a query__ that achieves the same end result. 
-    - Additionally, the same query might perform differently on different database engines (e.g., MySQL vs. PostgreSQL), and therefore there can be an infinite number of ways to structure a query that achieves the same end result. +  - **How to reduce the number of conditions?** The database administrator (DBA) or developer uses **heuristic methods** like: 
-  - **Reducing the Number of Solutions:** +    - Query profiling tools (e.g., EXPLAIN in SQL) to examine how different query structures perform. 
-    - The database administrator (DBA) or developer uses heuristic methods like: +    - Applying **best practices** like indexing the right columns, minimizing nested queries, and using joins effectively.
-      - Query profiling tools (e.g., EXPLAIN in SQL) to examine how different query structures perform. +
-      - Applying **best practices** like indexing the right columns, minimizing nested queries, and using joins effectively.+
     - By profiling and tweaking different versions of the query, the developer reduces the number of possible query structures to a few that perform optimally in the given context.     - By profiling and tweaking different versions of the query, the developer reduces the number of possible query structures to a few that perform optimally in the given context.
-    - There may not be a “single” optimal solution, but the best solution is found by reducing the possibilities through trial and error, experience, and tools. +    
 The //reductive approach// in database query optimization involves narrowing down many possible solutions (query structures) to a few practical ones. The solution can’t simply be inverted from the final result (i.e., retrieving the data); instead, developers use heuristics, profiling, and experience to eliminate inefficient options and find the most effective query structure for their specific environment. The //reductive approach// in database query optimization involves narrowing down many possible solutions (query structures) to a few practical ones. The solution can’t simply be inverted from the final result (i.e., retrieving the data); instead, developers use heuristics, profiling, and experience to eliminate inefficient options and find the most effective query structure for their specific environment.
tanszek/oktatas/techcomm/information_-_basics/sciences.1725915764.txt.gz · Last modified: by knehez