{"corpus":"steiner","locus":"GA 125/1910-08-26/p015","lang":"en","edition":"English","edition_basis":null,"is_original":false,"is_default_edition":true,"text":"Now I do not want to get into a discussion here of certain contributions – or the characteristics of these – that can be applied to the field of knowledge from a mathematical point of view. I will only point out the symptomatic nature of these things, pointing out that it is precisely in the field that has retained its wonderful inner strength, in the mathematical field, that it has become most apparent how the thinking of the 19th century has matured to the point of bursting the shell that seals off the human subject of knowledge from the transcendental world. And even if they are only hypotheses, sometimes boldly asserted, which have been pursued purely mathematically, we must take what has happened in the mathematical field as an expression of the longing of human knowledge to go beyond the sensual world. And there we have seen this longing realized precisely in the mathematical field. After all, mathematics in its forms, where it is called geometry, has held certain things to be unshakable since ancient Euclid! Who could have believed, for example, that there is anything more unshakable than the theorem that the three angles of a triangle equal 180 degrees, or the other theorem that if you have a straight line here and a point next to it, then in the sense of Euclidean geometry you can only draw a single parallel to the straight line through this point. That is, in the sense of Euclidean geometry, the sum of the angles of a triangle is equal to 180 degrees, and in its sense, only one parallel can be drawn to this line. This follows from the assumptions of Euclidean geometry. Who could now believe that this could be different! Nevertheless, and this is the significant thing. As I said, I could say many things for and against the content, but I am only addressing the symptomatic aspect of it, the longing to get out of the sensual realm; I just want to characterize it. The strange thing is that in the 19th century we saw the emergence of geometries other than Euclidean geometry. So it tried to crystallize out of itself a geometry — or geometries — that would be valid apply to something other than our ordinary sensory space, because for this space it is true that the three angles of a triangle together add up to 180 degrees, and that through a point next to a straight line you can only draw a parallel. And in the 19th century, we acquired geometries that do not want to apply only to our sensory space, no. Riemannian and Lobachevskyan geometry are two real geometries, born out of human thought according to strict mathematical laws.","text_trust":"verbatim","quote_safe":true,"title":"VI. The Present State of Philosophy and Science","site_heading":null,"date":"1910-08-26","location":null,"word_count":456,"url":"https://anthroposophy.ai/v1/steiner/en/ga/125/1910-08-26/p015","alignment_confidence":"high","alignment_reason":"beads:identity","two_voice_volume":false,"translation_credit":{"scope":"unit","status":"established","translator":"Steiner Online Library","identified_edition":"SOL2024","book_title":"Paths and Goals of Spiritual Man","publisher":"Steiner Online Library","publisher_inferred":true,"match_score":1.0,"source_url":"https://www.rsarchive.org/Lectures/GA125/English/SOL2024/19100826p01.html","evidence":"lecture_page_line","publisher_note":"The publisher is inferred from the edition code in this lecture's source URL, not stated by the source page. The translator IS verified for this lecture: its own source names the translator (evidence: the lecture's page, rsarchive's lecture list, or the cover of the book it was published in), and that page's text was matched against this unit (see match_score and source_url). Cite the translator with confidence; attribute the publisher only as probable."},"edition_coverage":{"de":["original"],"en":["English"]},"available_versions":[{"lang":"de","edition":"original","edition_basis":null,"is_original":true,"is_default_edition":true,"url":"https://anthroposophy.ai/v1/steiner/de/ga/125/1910-08-26/p015"},{"lang":"en","edition":"English","edition_basis":null,"is_original":false,"is_default_edition":true,"url":"https://anthroposophy.ai/v1/steiner/en/ga/125/1910-08-26/p015","paragraph_aligned":true}],"record":{"record_type":"unknown","scope":"volume","note":"How this text was taken down (shorthand transcript, listeners' notes or manuscript) is not on record here. Unknown is not an assurance of a stenogram: check the edition's own notes before resting an argument on one word.","evidence":null,"evidence_translation":null,"evidence_source":null},"paired_german":{"status":"paired","german":[{"locus":"GA 125/1910-08-26/p015","passage_number":15,"text":"Nun möchte ich mich hier nicht auf gewisse Beiträge einlassen — auf die Charakteristik derselben -, die gerade vom mathematischen Standpunkt aus auch auf das Erkenntnisgebiet gebracht werden können. Ich will nur auf das Symptomatische dieser Dinge hinweisen, darauf hinweisen, daß gerade auf dem Gebiet, das sich seine wunderbare innere Stärke bewahrt hat, auf dem mathematischen Gebiete, am klarsten hervorgetreten ist, wie das Denken des 19. Jahrhunderts durch sich selbst reif ist, die Hülle zu sprengen, welche das menschliche Erkenntnissubjekt gegenüber der übersinnlichen Welt abschließt. Und wenn es auch nur Hypothesen sind, manchmal waghalsig behauptet, die rein rechnerisch betrieben worden sind, so müssen wir das, was auf mathematischem Gebiet geschehen ist, doch so nehmen, daß es ein Ausdruck ist für die Sehnsucht des menschlichen Erkennens, hinaus aus der sinnlichen Welt. Und da haben wir gerade auf mathematischem Gebiet diese Sehnsucht sich verwirklichen sehen. Hat doch überhaupt die Mathematik in ihren Formen, wo man sie Geometrie nennt, seit dem alten Euklid gewisse Dinge für unerschütterlich gehalten! Wer konnte denn zum Beispiel glauben, daß es irgend etwas Unerschütterlicheres gibt als den Satz: Die drei Winkel des Dreiecks sind gleich 180 Grad -, oder den anderen Satz: Wenn Sie hier eine gerade Linie haben und neben dieser einen Punkt, so können Sie im Sinne der Euklidischen Geometrie durch diesen Punkt nur eine einzige Parallele zu der Geraden ziehen. — Das heißt im Sinne der Euklidischen Geometrie ist die Summe der Winkel eines Dreiecks gleich 180 Grad, und in ihrem Sinn kann man nur eine einzige Parallele zu dieser Geraden ziehen. Das folgt aus den Voraussetzungen der Euklidischen Geometrie. Wer könnte nun glauben, daß das anders sein könnte! Dennoch, und das ist das Bedeutsame — wie gesagt, ich könnte gegen das Inhaltliche manches Pro und Kontra sagen, aber ich gehe nur auf das Symptomatische daran ein, auf die Sehnsucht: heraus aus dem sinnlichen Gebiet; ich will nur charakterisieren -, da ist das Eigentümliche, daß wir im 19. Jahrhundert andere Geometrien haben heraufsteigen sehen, als die Euklidische ist. Also es versuchte die innere Schärfe des Denkens mit dem, was man denkerisch bei der Herausmeißelung geometrischer Wahrheiten zugrunde legt, es versuchte dieses Denken aus sich selber heraus zu kristallisieren eine Geometrie — oder Geometrien, die für etwas anderes gelten als für unseren gewöhnlichen Sinnesraum, denn für diesen gilt es, daß man die drei Winkel eines Dreiecks als zusammen 180 Grad ansprechen muß, daß man durch einen Punkt neben einer Geraden nur eine Parallele ziehen kann. Und wir haben im 19. Jahrhundert Geometrien bekommen, die nicht für unseren Sinnesraum nur gelten wollen, nein. Die Riemannsche und die Lobatschewskijsche Geometrie sind zwei wirkliche Geometrien, herausgeboren aus dem menschlichen Denken nach strengen mathematischen Gesetzen.","text_trust":"verbatim","quote_safe":true,"word_count":445,"url":"https://anthroposophy.ai/v1/steiner/de/ga/125/1910-08-26/p015"}],"english":["GA 125/1910-08-26/p015"],"method":"identity","confidence":"high","correspondence":"established","length_check":{"de_words":445,"en_words":456,"ratio":1.02,"partial":false},"pairing_state":"verified","verified":true,"evidence":["length","meaning","anchors","audit"]},"witness_completeness":{"ratio":1.04,"band":"comparable","note":null,"word_count":7932,"original_word_count":7595,"may_be_abridged":false},"witness_quality":{"translator":{"scope":"unit","status":"recorded","translator":"Steiner Online Library","identified_edition":"SOL2024","book_title":"Paths and Goals of Spiritual Man","publisher":"Steiner Online Library","publisher_inferred":true,"match_score":1.0,"source_url":"https://www.rsarchive.org/Lectures/GA125/English/SOL2024/19100826p01.html","evidence":"lecture_page_line","publisher_note":"The publisher is inferred from the edition code in this lecture's source URL, not stated by the source page. 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