Concept of Ganesa
Śhiva is process of perception and source of knowledge. It is the root desire of Brahma which started creation. In space, it is perception of hot and calm zones called Rudra (within 100 sun diameters of sun), Siva (100 to 1000 diameters), Sivatara (1000 to 10,000 diameters), Sivatama (104 to 105 diameters). Beyond that is Sadāśiva which is with Ambha (water like spread of interstellar matter in galaxy), son of it is Sāmba- Sadāśiva. Within the human body, it is capability of perception.
These are 3 pādas of Gāyatrī mantra. Cause or source of perception is Śiva, its effect is Hanumān. Literally it means having 2 lips (Hanu). Junction of 2 lips is also the junction of karmendriya (organs of action) and Jñānendriya (organs of sense). So the person having complete harmony of action and wisdom is called Hanumān. First part of Gāyatrī mantra is creation process after initial desire of Brahma. The creation was from ocean of Rasa like rain drops from cloud-called Vṛṣā (causing rains, bull-both are inputs of the basic yajña of farming). Creation was always like the previous one as rules are the same-so it is called Kapi = monkey which has habit of copying human actions. So, Kapi = copy. Creator aspect is Vṛṣā-kapi, name of Hanumān in Vedas. Physical aspect indicated by part 2 of Gāyatrī mantra is motion caused by temperature difference, whose perception is Śiva. That is called Māruti, i.e. born of Marut = wind. Source and perception of knowledge is part 3 of Gāyatrī mantra. Its effect is Hanumān. Guru is Śiva, disciple or his improvement is Hanumān.
Chain of Guru-śiṣya is indicated by Bunyan tree which is symbol of Śiva. An aerial root descending from Bunyan branch grows into a similar tree. Similarly, guru makes a man like himself by imparting knowledge to śiṣya. Central bunyan is Śiva, secondary trees are Hanumān. These are Druma (tree) from central Druma, so the complex is called Dumduma. There are Dumduma around places of Śiva-around Lingarāja in Bhubaneswar, around Dakṣiṇeśvara in Kolkata (Dumdum airport), near Kāmākhyā in Asam, Dumka near Baidyanath-dham in Jharkhand, Dumiyani near Somnath in Gujrat, and Damdami Taksal of Haramandir in Amritsar.
Knowledge is an ordered structure. Any process creates more disorder, called entropy in second law of thermodynamics. If we acquire ordered knowledge, it creates more disorder in surroundings. By observing any object for knowing it, we disturb it. So, Śiva is Pralaya (deluge) aspect. Dance of creation is Rāsa (harmony) or Lāsya. Dance of destruction is Tāṇḍava (disorder, havoc) of Śiva. In human form, there were 28 incarnations of Śiva from Svāyambhuva Manu (Brahmā) in 29100 BC to Vedavyāsa in 3200 BC. They were sources of knowledge. In between many incarnations were crucial for destruction of asuras. There were 9 incarnations of Hanumān described in Parāśara samhitā.
After acquiring, knowledge is structured in 2 ways-visible and separate entities can be counted. Anything which can be counted (Gaṇanā) is Gaṇeśa. Field of space or continuous matter is like Rasa (homogenous). Its knowledge is Sarasvatī (Sa + Rasa + vatī = That is with Rasa). These are opposite ends of knowledge called-countable & abstract nouns, Bīja & Kṣetra Gaṇita (discrete, field theories), Digital & analog. After I explained this concept to Prof Abul Salam in 1979, he wrote an article or comment that controller of digital computer (Gaṇeśa should be called mouse-his vehicle. Within the cobweb of 100 billion neurons of mind, our sensation moves as mouse moves in earth, so vehicle of Gaṇeśa is mouse. Central part of human brain is like elephant head, called Gaṇeśa guhā in tantra. Its trunk descends as Suṣumnā having 6 chakras till end of vertebral column.
Representation of knowledge in letters, words, sentence is countable and is called Gaṇeśa. Its meaning and sense is Sarasvatī is not countable. Both have been prayed in this manner by Tulasidas in his Hindi epic Rāma-charita-mānasa-
वर्णानामर्थ संघानां रसानां छन्दसामपि। मंगलानां च कर्त्तारौ वन्दे वाणी विनायकौ॥ (रामचरितमानस, मंगलाचरण)
Thus all visible entities are Gaṇeśa as stated in his Atharva-śīrṣa
त्वमेव प्रत्यक्षं ब्रह्मा असि (गणपति अथर्व शीर्ष)
These indicate 3 types of infinity described in Cantor’s set theory-countable with natural numbers (samkhyeya), rational numbers (asamkhyeya), and beyond expression by algebraic equations (Aparimeya). These names are in Viṣṇu-sahasranāma-Ananta, asamkhyeya, aprameya. Collection of words is Gaṇeśa (Pratyakṣa Brahma) or infinity of natural numbers. Its meaning is in between discrete words, called Subrahma (Gopatha Brāhmaṇa, Pūrva 1/1) = Brahma parts united by Sveda (water, sweating). This is like fuzzing of separate dots of ink sprayed by ink jet printer or pattern recognition by collection of dots which look separate in magnification or zooming. Physical form is called Subrahmaṇya (Kārttikeya)-another son of human Śiva. Feelings in mind are Sarasvatī. ॐ ब्रह्म ह वा इदमग्र आसीत्... तस्य श्रान्तस्य तप्तस्य सन्तप्तस्य ललाटे स्नेहो यदार्द्र्यमजायत। महद् वै यक्षं सुवेदं अविदामह इति। .. एतं सुवेदं सन्तं स्वेद इति आचक्षते॥१॥ सर्वेभ्यो रोमगर्त्तेभ्यः स्वेदधाराः प्रस्यन्दन्त। ... अहं इदं सर्वं धारयिष्यामि.... तस्मात् धारा अभवन्॥२॥ (गोपथ ब्राह्मण पूर्व १/१-२)
In the beginning, there was Brahma only. By work and heat, sweat flowed from his forehead. That suveda was called sveda and it held (joined) roma-garttas (hair-roots, point objects). Thus, Brahma became Subrahma. The 3 types of infinity are also indicated in Īśāvāsya Upaniṣad-
स पर्यगात् शुक्रम् अकायम् अव्रणम् अस्नाविरम् शुद्धम् अपापविद्धम्। कविर्मनीषी परिभूः स्वयम्भूः याथातथ्यतोऽर्थान् व्यदधात् शाश्वतीभ्यः समाभ्यः। (ईशावास्य उपनिषद् ८, वाज. यजु. ४०/८)
Abstract source is without form, defects, links etc. By closing it in boundary, forms links etc are created (in world, word also). The person who transforms the inner thoughts perfectly in words (or creates exact replica of abstract) makes eternal creation (poetry, or man etc as image of Brahma). Here, discrete objects are natural numbers, their links etc are rational numbers and the abstract source or field are transcendental numbers.
Worship of Gaṇeśa and Sarasvatī are at 2 ends of year cycle, separated by 6 months on Bhādra śukla 4 and Māgha śukla 5.
The first person deputed by Brahmā to create script was called Gaṇapati, who used ṛṇa (minus sign, dash) & chid-ṛṇa (small part of that, dot) to write letters. He used 3 pairs of dash-dot (like I-ching of China) to write symbols of Devas as letters. With 3 pairs, 26 = 64 letters are possible in Brāhmī script.
गणानां त्वा गणपतिं हवामहे, कविं कवीनामुपमश्रवस्तम्। ज्येष्ठराजं ब्रह्मणा ब्रह्मणस्पत आ नः शृण्वन्नृतिभिः सीद सादनम्॥१॥ विश्वेभ्यो हित्वा भुवनेभ्यस्परि त्वष्टाजनत् साम्नः कविः। स ऋणया चिदृणया ब्रह्मणस्पतिर्द्रुहो हन्तमह ऋतस्य धर्तरि॥१७॥ (ऋक् २/२३/१, १७) देवलक्ष्मं वै त्र्यालिखिता तामुत्तर लक्ष्माण देवा उपादधत.... (तैत्तिरीय संहिता ५/२/८/३)
Population of a country is also countable and is called Gaṇa. Its head is Gaṇapati = President. Servants of head of state steal despite any amount of vigilance. That aspect is indicated by mouse. Sanskrit word Mūṣik means thief. State runs by collection of tax from people, indicated by elephant drawing water through its trunk. Hand of man is ‘Kara; as it does work. For elephant, its trunk does all work, hence it is also ‘Kara’. Similar to drawing water by this ‘Kara’, tax is collected from people, so tax also is ‘Kara’. Gaṇeśa also is praised as Karāṭa (collector of tax) which is also a title in south India.
तत् कराटाय च विद्द्महे हस्तिमुखाय धीमहि , तन्नो दन्ती प्रचोदयात् । (कृष्णयजुर्वेदः - मैत्रायणीशाखा - अग्निचित्प्रकरणम् -119 - 120)
There were many Gaṇapatis in human form called Bṛhaspati, Angirā etc who designed scripts in various eras.