{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "9VW7VX0SXB"
    }
  ],
  "body": "\n## Definition\n\n*Constructivism* is the position in the philosophy of mathematics that consists of these two principles.\n\n1. A mathematical object exists only if there is a method to construct it.\n2. A proof of \"there exists $x$ such that $P(x)$\" must give a method to construct such an $x$. It must also prove $P(x)$ for this $x$.\n\n*Constructive mathematics* is mathematics carried out under these principles.\nIts logic is intuitionistic logic: classical logic without the general [[law of excluded middle|0RYYK8PG4S]] $P \\vee \\neg P$.\n\n## Remarks\n\nExample: the proof that for each natural number $n$ there is a prime $p \u003e n$, by taking a prime factor of $n! + 1$, is constructive.\nIt gives a method: factor $n! + 1$ and take any prime factor $p$; then $p \u003e n$, because no number from $2$ to $n$ divides $n! + 1$.\nNon-example: the classical proof that there are irrational numbers $a, b$ with $a^b$ rational is not constructive.\nIt argues that either $\\sqrt{2}^{\\sqrt{2}}$ is rational, or it is not and then $(\\sqrt{2}^{\\sqrt{2}})^{\\sqrt{2}} = 2$; it does not say which case holds.\nIt uses the law of excluded middle, which fails principle 2.\nIn constructive mathematics, a [[proof by contradiction|0J0XBKG64S]] proves negated statements $\\neg P$, but in general does not prove $P$ from $\\neg \\neg P$; see the [[double negation law|0NFAGWH84K]].\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "G958EQ8CWW",
    "kind": "definition",
    "msc": [
      "03F65",
      "03F55",
      "00A30"
    ],
    "name": "Constructivism",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics)"
      }
    ],
    "tag": "F1P5SZ1XPH",
    "uses": [
      "01XM7JEKHY",
      "0RYYK8PG4S",
      "0J0XBKG64S",
      "0NFAGWH84K"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/F1P5SZ1XPH/",
  "links": [
    {
      "field": "defines",
      "target": "G958EQ8CWW"
    },
    {
      "field": "uses",
      "target": "01XM7JEKHY"
    },
    {
      "field": "uses",
      "target": "0RYYK8PG4S"
    },
    {
      "field": "uses",
      "target": "0J0XBKG64S"
    },
    {
      "field": "uses",
      "target": "0NFAGWH84K"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/F1P5SZ1XPH.md",
  "msc": [
    "03F65",
    "03F55",
    "00A30"
  ],
  "name": "Constructivism",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/F1P/F1P5SZ1XPH.md",
  "status": null,
  "tag": "F1P5SZ1XPH",
  "uri": "m2://F1P5SZ1XPH",
  "url": "https://math2.tamnd.com/wiki/constructivism-definition/",
  "writing": {
    "tag": "F1P5SZ1XPH",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}