{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\n*Intuitionism* is the position in the [[foundations of mathematics|EFFRJFW67Y]] that consists of these theses:\n\n1. mathematical objects are mental constructions, and an object exists only if it can be constructed;\n2. a mathematical statement is true only if it has a [[constructive proof|01XM7JEKHY]], so truth means provability by construction;\n3. as a consequence, the law of excluded middle $P \\lor \\neg P$ is not accepted for arbitrary statements $P$.\n\n## Remarks\n\nThe statement \"there exist irrational numbers $a$ and $b$ with $a^b$ rational\" has an intuitionistic proof.\nTake $a = \\sqrt 2$ and $b = 2 \\log_2 3$; then $a^b = 2^{\\log_2 3} = 3$.\nBoth numbers are shown to be irrational directly.\nThe classical proof considers the two cases \"$\\sqrt 2^{\\sqrt 2}$ is rational\" and \"it is irrational\".\nThis proof is not intuitionistic: it uses excluded middle and does not say which case holds.\nThe logic of intuitionism was formalized by Arend Heyting as [[intuitionistic logic|Q11N8X99FK]], and its meaning of the connectives is the [[Brouwer–Heyting–Kolmogorov interpretation|7B53ZGXXEQ]].\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "XQHF63BBYF",
    "kind": "definition",
    "msc": [
      "03F55",
      "03A05"
    ],
    "name": "Intuitionism",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Intuitionism"
      }
    ],
    "tag": "RR10R8JN02",
    "uses": [
      "EFFRJFW67Y",
      "01XM7JEKHY"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/RR10R8JN02/",
  "links": [
    {
      "field": "defines",
      "target": "XQHF63BBYF"
    },
    {
      "field": "uses",
      "target": "EFFRJFW67Y"
    },
    {
      "field": "uses",
      "target": "01XM7JEKHY"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/RR10R8JN02.md",
  "msc": [
    "03F55",
    "03A05"
  ],
  "name": "Intuitionism",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RR1/RR10R8JN02.md",
  "status": null,
  "tag": "RR10R8JN02",
  "uri": "m2://RR10R8JN02",
  "url": "https://math2.tamnd.com/wiki/intuitionism-definition/",
  "writing": {
    "tag": "RR10R8JN02",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}