tanszek:oktatas:techcomm:information_-_basics:sciences
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tanszek:oktatas:techcomm:information_-_basics:sciences [2024/09/09 21:13] – [Reductive Sciences] knehez | tanszek:oktatas:techcomm:information_-_basics:sciences [2025/09/08 19:42] (current) – [Deductive 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//, // | + | According to the definition: //Science// is understood as a provable and fact-based system of the objective relationships between //nature//, // |
- | //Science// is not just a collection of knowledge, but a discovery process. //Science// aims to discover new information, | + | //Science// is not just a collection of knowledge, but a **discovery process**. //Science// aims to discover new information, |
- | //Science// is distinguished from other historically established forms of social consciousness by the following characteristics: | + | Science, among our historically established forms of social consciousness, is distinguished and emphasized |
- | + | ||
- | //Science// has been highlighted because of the following criteria from our historically established social forms of consciousness: | + | |
* they possess high-reaching concepts or logical tools to formulate or express broad, general or universal **principles** or **laws** (e.g. gravity, axioms, [[https:// | * they possess high-reaching concepts or logical tools to formulate or express broad, general or universal **principles** or **laws** (e.g. gravity, axioms, [[https:// | ||
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According to **principles**, | According to **principles**, | ||
+ | |||
+ | ---- | ||
====== Inductive Sciences ====== | ====== Inductive Sciences ====== | ||
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**Explanation: | **Explanation: | ||
- | 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:// | + | 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:// |
- | 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**: | + | ---- |
+ | |||
+ | **Example | ||
**Problem**: | **Problem**: | ||
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- **Inductive Hypothesis**: | - **Inductive Hypothesis**: | ||
- **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 \(n = k\): | ||
+ | |||
+ | $$ 1 + 2 + \cdots + k = \frac{k(k+1)}{2}. $$ | ||
+ | |||
+ | **Simplify**: | ||
+ | $$ \frac{k(k+1) + 2(k+1)}{2} = \frac{(k+1)(k+2)}{2}. $$ | ||
+ | |||
+ | Thus, the formula also holds for \( n = k+1 \). | ||
https:// | https:// | ||
+ | |||
+ | ---- | ||
====== 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 – a famous Hungarian mathematician – wrote this famous sentence to his father: | ||
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{{: | {{: | ||
+ | |||
+ | The quote from [[https:// | ||
+ | |||
+ | ---- | ||
====== Reductive Sciences ====== | ====== Reductive Sciences ====== | ||
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- | **Explanation**: | + | **Explanation**: |
- | 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 |
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- 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 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: |
- | | + | |
- **There’s no single “perfect” solution** | - **There’s no single “perfect” solution** | ||
- Additionally, | - Additionally, | ||
- | - **Reducing the Number of Solutions:** | + | - **Reducing the Number of Conditions:**: The database administrator (DBA) or developer uses **heuristic methods** like: |
- | - The database administrator (DBA) or developer uses **heuristic methods** like: | + | - Query profiling tools (e.g., EXPLAIN in SQL) to examine how different query structures perform. |
- | - 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. |
- | - 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. | ||
- | | + | |
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.1725916399.txt.gz · Last modified: 2024/09/09 21:13 by knehez